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Theory of Stochastic Laplacian Growth

Oleg Alekseev and Mark Mineev-WeinsteinInternational Institute of Physics,

Federal University of Rio Grande do Norte,

59078-400, Natal, Brazil

(Dated: November 29, 2018)

We generalize the diffusion-limited aggregation by issuing many randomly-walking particles, whichstick to a cluster at the discrete time unit providing its growth. Using simple combinatorial argu-ments we determine probabilities of different growth scenarios and prove that the most probable evo-lution is governed by the deterministic Laplacian growth equation. A potential-theoretical analysisof the growth probabilities reveals connections with the tau-function of the integrable dispersionlesslimit of the two-dimensional Toda hierarchy, normal matrix ensembles, and the two-dimensionalDyson gas confined in a non-uniform magnetic field. We introduce the time-dependent Hamilto-nian, which generates transitions between different classes of equivalence of closed curves, and provethe Hamiltonian structure of the interface dynamics. Finally, we propose a relation between prob-abilities of growth scenarios and the semi-classical limit of certain correlation functions of “light”exponential operators in the Liouville conformal field theory on a pseudosphere.

I. INTRODUCTION

The problem of pattern formation in non-equilibriumgrowth processes has attracted great interest since the1980s. These processes can be conventionally dividedinto different classes depending on the physical mecha-nisms governing the dynamics. The growth occurring insystems where diffusion plays an important role deservesparticular interest, e.g. the formation of river networks,frost on glasses, growth of bacterial colonies, electricbreakdown, dendritic crystals in rocks, polymerization,and many others. The patterns occurring in these sys-tems are in general scale-invariant multi-branched fractalclusters, and the multi-fractal analysis can be applied todescribe their different universality classes. Remarkably,the complex shapes, which are related to the instabilityof the diffusion-limited process, are universal, i.e. theirlong-range properties depend only slightly on the detailsof the interaction between particles. They have somegeneral features providing a possibility to get insight intocomplex pattern formation by means of simple aggrega-tion models. The best known example is the diffusion-limited aggregation (DLA), proposed by T. Witten andL. Sander in 1981 [1]. Due to the universality this modelis widely applicable to study formation of complex pat-terns and their fractal properties in non-equilibrium andhighly unstable growth processes. However, despite itssimplicity many aspects of DLA remain puzzling.

In short, DLA describes aggregation of small parti-cles undergoing Brownian motion to form clusters. Thedensity of the particles is assumed to be low, so thatthe aggregation process occurs by one particle per timeunit. This process can be described by the followingmodel. Consider a seed particle at the origin of a two-dimensional lattice. Another particle is launched from adistant source and is allowed to walk randomly. Eventu-ally the second particle will stick irreversibly to the seed

thus occupying one of lattice sites adjacent to the origin.Then the third randomly walking particle is launchedfrom the source. It moves around the lattice until it sticksto the two-particle cluster, and so forth. This processleads to a highly-branched fractal cluster. Remarkablyall clusters grown in this way are mono-fractals with thesame Hausdorff (fractal) dimension, dF = 1.71 ± 0.01,obtained numerically, which appears to be robust anduniversal [2]. This number does not depend on the lat-tice structure, depends weakly on the geometry of theproblem, and still is out of analytical reach.

Let us briefly recall a mathematical description ofDLA. The growth of the cluster is specified by the set ofgrowth probabilities, i.e. probabilities that some perime-ter site is next to be added to the cluster. To be more spe-cific, let u(x, t) be the probability that the random walkerreaches the point x at time t. Assuming that the growthis sufficiently slow, such that ∂u/∂t can be neglected,the probability satisfies the Laplace equation, ∇2u = 0,with a condition u = 0 at the boundary of the cluster.The growth velocity of the interface is proportional to thegradient of the probability density at the boundary, ∇nu,where n is the unit normal. Remarkably, the same set ofequations describes a completely different (at first sight)non-equilibrium process, namely, the Laplacian growthin a Hele-Shaw cell [3]. In the Laplacian growth prob-lem the pressure field satisfies the Laplace equation withthe constant pressure at the boundary (if to neglect sur-face tension), while the velocity of the interface is pro-portional to the pressure gradient. More precisely, theprobability u(x, t) is equivalent to (minus) pressure. Itshould be noted that DLA corresponds to unstable Lapla-cian growth so that the inviscid fluid pushes the viscousone. The latter problem is ill-posed in a mathematicalsense, because the evolution of an arbitrary interface of-ten leads to generation of cusps within a finite time [4].To handle this problem one can use a regularization, such

2

as surface tension. However, a wide class of non-singularlogarithmic exact solutions are known to exist even forzero surface tension [5, 6].

Another remarkable feature of the Laplacian growthproblem is the formation of universal asymptotic shapesobserved in experiments. The best known example is theSaffman-Taylor finger propagating in a long rectangularHele-Shaw cell [7]. A continuous family of possible fin-gers labeled by λ (the ratio of the finger width to thechannel width) can be easily obtained [7]. However, ex-perimentally only the unique pattern with λ = 1/2 isobserved. This fact raised a non-trivial problem of se-lection of a single pattern from infinitely many solutions.Historically, the problem was addressed first by includ-ing surface tension and applying the WKB-like technique(”the asymptotic beyond all orders”) to study the low-surface-tension limit [8, 9]. More recently it was shownthat the selection problem can be addressed by exactsolutions of Laplacian growth even without surface ten-sion [10]. The analysis was based on the deterministictime-dependent solutions, which implies that the shapeof the initial interface is specified and evolves with timedeterministically.

However, the Laplacian growth is known to be ahighly unstable, dissipative, non-equilibrium, and non-linear phenomenon. It is expected that the shape ofthe interface can change stochastically during the evolu-tion. Therefore, it seems reasonable to consider stochas-tic Laplacian growth, and its deep connection with DLAprovides a clue to the problem. The well-known attemptto address the problem from this point of view was madeby M. Hastings and L. Levitov by means of iteratingstochastic conformal mappings [11]. In their approachthe cluster is grown by adding a small semicircular pieceto the interface with a probability, which depends on thelocal electric field that is a gradient of pressure in termsof the Laplacian growth. In the continuum limit, wherethe attached bumps are infinitesimally small, the growthis described by the deterministic Laplacian growth equa-tion.In this paper we develop another approach to the

stochastic pattern formation in the Laplacian growth,which allows one to study the grown clusters analytically.The model we propose is simple and serves as a bridgebetween the Laplacian growth and DLA. It is assumedthat K particles (instead of one) are deposited from adistant source per time unit. The particles are assumedto be uncorrelated and move around until they stick tothe cluster interface. As soon as all of them become apart of the cluster other K particles are released from adistant source, and the process continues in this way.

For K = 1 the model is equivalent to the ordinaryDLA, while for large K it describes the stochastic Lapla-cian growth. In the limit, when the particles’ sizes goto zero, we recover the deterministic Laplacian growthdynamics, governed by a single hydrodynamical source

at infinity. It is convenient to consider roughly square-shaped particles, whose sides are formed by segments ofthe equipotential and field lines. Being stuck to the in-terface each time step, they form an external layer of theemergent domain. Since their shapes are specified, theevolution of the cluster is completely determined by thedistributions of the attached particles along the domainboundary. Therefore a statistical description of differentgrowth processes becomes possible.The paper is organized as follows. In Section II we fix

the notation and review classical Laplacian growth andits integrable structure. In Section III we introduce thestochastic Laplacian growth model and define the proba-bility of the clusters. Besides, we prove that the classical(deterministic) limit of the model describes the determin-istic Laplacian growth governed by a single hydrodynam-ical source at infinity. In other words, we introduce thevariational approach to the Laplacian growth and deter-mine the action, which extremum gives the equation fordissipative motion of the boundary of the domain. So farthe Laplacian growth equation was derived only as theapproximation of viscous hydrodynamics. The rest partof the section is dedicated to other (non-classical) growthscenarios and their statistical weights, which are relatedwith the entropy (number of states with specific param-eters) of the growing clusters. Surprisingly, these purelyprobabilistic expressions can be transformed to the elec-trostatic energy of a specific charge distributions in theexterior of the growing domain. Thus, the statisticalweights of the clusters obey the usual Gibbs-Boltzmanndistribution, which, as a rule, is not applied out fromequilibrium. The relations between our results and sometopics of the modern mathematical physics, such as thetheory of random matrix and the Kahler geometry, arealso mentioned. In Section IV we prove the Hamilto-nian structure of the interface dynamics and interpretthe Laplacian growth equation as the Hamilton’s equa-tion for a certain dynamical system. In Section V we in-dicate an interesting connection between our results andthe Liouville conformal field theory. Finally, we draw ourconclusions.

II. CLASSICAL LAPLACIAN GROWTH

1. Standard formulation and conformal description

of the Laplacian growth

The Saffman-Taylor problem (known also as the Lapla-cain growth) describes motion of the interface betweentwo incompressible fluids with different viscosities in aHele-Shaw cell. Equations for interface dynamics takeextremely simple form in the limit when one fluid is in-viscid and surface tension is zero. For brevity, we willrefer to inviscid and viscous fluids as water and oil re-spectively. We also assume that the region occupied by

3

water, D+, is simply connected and contains the originof the two-dimensional plane, while the oil domain willbe denoted by D− ≡ D = C \ D+. In the water (withzero viscosity) the pressure field is a constant. In theoil domain, in turn, the fluid velocity is governed by theDarcy’s law (eq. (1) below). Let z = x + iy be a com-plex coordinate on the plane. Let v(z, t) and p(z, t) bethe velocity and the pressure field in the oil domain re-spectively. Then the Darcy’s law expresses a simple pro-portional relationship between v(z, t) and the pressuregradient:

v(z, t) = −gradp(z, t), z ∈ D, (1)

where we set to 1 the proportionality positive constant,which depends on viscosity and the thickness of the Hele-Shaw cell. In the absence of surface tension the pressureat the interface vanishes,

p(z, t)∣

∣

∣

∂D= 0. (2)

A continuity condition at the interface requires that thenormal boundary velocity equals to the normal fluid ve-locity, Vn(ξ, t), at the interface:

Vn(ξ, t) = −∂np(ξ, t), ξ ∈ ∂D, (3)

where ∂n is a normal derivative.Consider a set of sources with rates q1, q2 . . . , qM , lo-

cated at the points z1, z2 . . . , zM ∈ D in the oil domain.Near the sources the pressure diverges logarithmically.Taking into account that both fluids are incompressible,the velocity of the fluid satisfies the divergence-free con-dition, divv = 0, everywhere in the oil domain except thesource positions. Since the velocity in the oil is propor-tional to the gradient of pressure (1), the latter satisfiesthe Laplace equation:

∆ p(z, t) =

M∑

m=1

qmδ(2)(z − zm), z, zm ∈ D, (4)

where ∆ = 4∂∂ is a Laplace operator on the plane, andδ(2)(z) is a two-dimensional delta-function.The equations (2) and (4) completely specifies pressure

and therefore the boundary velocity in (3). The uniquesolution for p can be written as the linear combination ofthe Green’s functions of the exterior Dirichlet boundaryproblem [12],

p(z, t) =

M∑

m=1

qm2πGD(t)(z, zm). (5)

The Green’s function is symmetric in its argu-ments and satisfies the Laplace equation: ∆G(z, z′) =2πδ(2)(z − z′), where we omitted the domain’s label Dfor brevity. As a function of z it is harmonic everywhere

in D except the point, z = z′, where it diverges logarith-mically: G(z, z′) = log |z − z′| + reg, as z → z′. Also,G(z,∞) = − log |z|+reg, as z → ∞. Besides, the Green’sfunction equals zero identically, G(z, ξ) = 0, ξ ∈ ∂D, ifone of its arguments belongs to the domain’s boundary.The Green’s function has a simple electrostatic inter-

pretation. Suppose the boundary of the domain, ∂D, is agrounded conducting wire [41] and at some point, z ∈ D,there is a unit charge, exerting an electric field of forcederived from a logarithmic potential. It is convenient todecompose the Green’s function into two parts:

G(z, z′) = log |z − z′|+G−(z, z′). (6)

The first function at the right hand side, G+(z, z′) =log |z − z′|, is a potential at z′ created by the point-likecharge located at z. The second function, G−(z, z′), isa potential of countercharges induced on the domain’sboundary. The Green’s function can then be consideredas a potential of the total field so created.The Green’s function for any simply-connected domain

can be written explicitly using the Riemann mapping the-orem. According to the theorem, the external domain Dis conformally equivalent to the compliment B = C \B+

of the unit disk B+ = w ∈ C : |w| < 1. Let thecomplex analytic function,

z = f(w) : C \B → D, (7)

realize this correspondence. The map is specifieduniquely by virtue of the conditions: f(∞) = ∞ andf ′(∞) > 0. By w = w(z) we will denote the inversefunction to z(w). By using these notation, the Green’sfunction in D can be written as follows:

G(z, z′) = log

∣

∣

∣

∣

∣

w(z)− w(z′)

1− w(z)w(z′)

∣

∣

∣

∣

∣

, (8)

where bar stands for complex conjugation.The equation of motion of the interface follows from (3)

by virtue of the conformal map z = f(w), as the pres-sure is specified (5). The normal velocity can be writ-ten in terms of the conformal map as Vn = Im(V τ) =Im(ztzl), where τ = dz/|dz| is a unit tangent vector, andl =

∫

|dz| is an arc-length of the boundary. Changing aparametrization of the boundary from the arc-length tothe angle at the w-plane we recast Vn in the form:

Vn(eiφ, t) = |f ′(eiφ)|−1 Im

(

∂tf(eiφ)∂φf(eiφ))

, (9)

where eiφ = w(ξ) is a pre-image of the boundary.The normal boundary velocity should be equated to

the pressure gradient at the interface (3). From (8) wehave

∂nGD(t)(ξ, zm) = |f ′(eiφ)|−1 Re eiφ + ameiφ − am

, (10)

4

where am is an inverse pre-image of zm = f(1/am), sofrom (3), (5), (9), and the last equation we obtain

Im(

∂tf(φ, t)∂φf(φ, t))

=

M∑

m=1

qm2π

Re eiφ + ameiφ − am

. (11)

This equation is a famous Laplacian growth equationwhich describes the interface evolution in the presenceof several oil sources [26]. In a case of a single source atinfinity, it takes a familiar form

Im(

∂tf(φ, t)∂φf(φ, t))

=q

2π, (12)

which was introduced in [27, 28].Remarkably, the non-linear LG equation admits differ-

ent classes of exact solutions. The simplest examples arepolynomial and rational solutions which, however, giverise to a cusp-like singularity of the interface in a finitetime [4]. This signifies importance of surface tension ef-fects at highly curved parts of the interface. Singularityfree interface dynamics with zero surface tension is de-scribed by logarithmic solutions (and even more generalmulti-cut ones [29]).Another remarkable property of the Laplacian growth

is an existence of an infinite set of conservation laws dis-covered by S. Richardson in 1972 [13]. Introducing theset of internal and external harmonic moments, vk andtk respectively, as follows:

vk =1

π

∫

D+

zkd2z, v0 =2

π

∫

D+

log |z|d2z,

tk = − 1

πk

∫

D

z−kd2z, t0 =1

π

∫

D+

d2z,

(13)

the Richardson’s theorem relates the time evolution of tkwith the locations and rates of the sources in D:

∂tk∂t

=1

π

M∑

m=1

qmz−km

k,

∂t0∂t

=1

π

M∑

m=1

qm, (14)

In particular, for the LG with a single source at infin-ity (12) all tk (k > 0) are constants, while the domain’sarea, πt0, growth linearly with time. If the sources ratesin (14) are constants, the time derivatives of tk are con-served. Thus, the Laplacian growth possesses an infinitenumber of integrals of motion, which is a distinct featureof integrability.The differential equations (14) can be readily inte-

grated. Therefore in order to determine the shape ofthe domain it is sufficient to reconstruct it from the setof external harmonic moments, tk and t0. This is a wellknown classical inverse potential problem of reconstruct-ing the domain from its Newtonian potential outside [14].Suppose the water domain, D+, is filled homoge-

neously with unit density. The Newtonian potential cre-ated by the domain is

Φ(z, z) = − 2

π

∫

D+

d2z′ log |z − z′|. (15)

Hereafter we will denote Φ(z, z) as Φ(z) for brevity. Thepotential (15) obeys the equation

−∆Φ(z) =

1, z ∈ D+,

0, z ∈ D− \∞,(16)

and admits the Taylor series expansions in the exteriorand interior domains respectively:

Φ−(z) = −2t0 log |z|+ 2Re∑

k>0

vkkz−k, (z ∈ D−),

Φ+(z) = −|z|2 − v0 + 2Re∑

k>0

tkzk, (z ∈ D+).

(17)Here vk and tk are the harmonic moments introducedin (13), thus v0 = −Φ(0) is the (minus) value of thepotential at the origin.The continuity of the potential and its gradient at the

interface is expressed by the conditions:

Φ+(z) = Φ−(z), ∂zΦ+(z) = ∂zΦ

−(z), z ∈ ∂D. (18)

This equations allow to reconstruct Φ+(z) from Φ−(z)(or vice versa) and, therefore, to treat the internal har-monic moments, vk (k > 0) and t0 as functions of ex-ternal ones, tk (k > 0), and v0. However, it is moreconvenient to chose t0 instead of v0 as an independentparameter. Therefore, any functional on the domain, e.gthe potential (15), can be considered as a function of theexternal harmonic moments, tk and tk, and the area πt0.Under certain conditions they completely determine theform of the curve as well as the moments vk (k ≥ 0).

2. Schwarz function description of the Laplacian

growth

It is known that the LG equation (11) can be integratedin terms of the Schwarz function [15], which turns out tobe of great use in different contexts. To introduce theSchwarz function of a certain contour Γ, defined by theequation F (x, y) = 0, we substitute x = (z + z)/2 andy = (z− z)/2, and solve the resulting equation locally forz in terms of z:

z = S(z). (19)

Then, the Schwarz function is defined as the analyticcontinuation of S(z) away from the curve. The confor-mal map and the Schwarz function are connected in thefollowing way:

z = f(w), S = f(1/w). (20)

The Herglotz’s theorem [16], which follows from theserelations, establishes a correspondence between singular-ities of the Schwarz function and of the conformal map.

5

Let a be a singular point of f(w) inside the unit circle,which appears with the coefficient A. Then the Schwarzfunction has a singularity of the same kind at b ∈ D witha coefficient B, which is related to (a,A) as:

b = f(1/a), B = A(−a2f ′(1/a))m, (21)

where m is a multiplicity of the pole, rational number orzero if a is a pole, algebraic branch point or logarithmicsingularity respectively.For any analytic contour the Schwarz function is well

defined in the strip-like neighborhood of the curve andcan be decomposed into the sum S(z) = S+(z) + S−(z),where the functions S±(z) are regular in D± respectively.Their expansions near the origin and infinity have a form:

S+(z) =∞∑

k=1

ktkzk−1, S−(z) =

t0z+

∞∑

k=1

vkz−k−1. (22)

The coefficients in these series expansions are nothingbut the harmonic moments introduced earlier, as readilyfollows from the Stokes theorem and (19):

∫

D

z±kd2z = ±∮

∂D

z±kzdz = ±∮

∂D

z±kS(z)dz. (23)

If the external harmonic moments, tk, in the Laplaciangrowth change according to (14), the time evolution ofS+(z) reads:

∂

∂tS+(z, t) = − 1

π

M∑

m=1

qmz − zm

. (24)

Thus, singularities of S+(z, t) are in one-to-one corre-spondence with the operating sources. The initial con-dition to (24), i.e. the function S+(z, 0), also can havesingularities, which can be treated as the frozen sources,operating during t < 0. In the internal domain theSchwarz function has a more complicated set of singu-larities. Typically, they are branch points of order twowith cuts between them.The Darcy’s law (1) relates the time evolution of the

Schwarz function with pressure in D. From z = S(z, t),by means of the chain rule we obtain:

V = S + S′V, (25)

where V = z. Introducing the unit tangent and normalvectors as τ = dz/|dz| and n = −iτ = 1/i

√S′ respec-

tively, where |dz| =√S′dz is an arc-length, the vanishing

of the tangential velocity of the interface can be writtenin the form: Vτ = Im(V n) = 0. Since S′ = 1/S′ wetransform the latter condition to V +S′V = 0. Togetherwith (25) it results in the relation: V = S/2. The Darcy’slaw can be also written as V = −2∂p, or, equivalently,V = −2∂p. Thus, we obtain:

S(z, t) = 2W ′(z, t), (26)

where dot and prime denote the time and space partialderivatives respectively, andW (z, t) is the so-called com-

plex potential, such that p = −ReW . One can referto (26) as to the Laplacian growth equation. Indeed, thenormal boundary velocity, Vn = Re(V n), can be writtenin terms of the Schwarz function:

Vn(ξ, t) =S(ξ, t)

2i√

S′(ξ, t), ξ ∈ ∂D. (27)

Being projected on the unit normal vector, the LG equa-tion (26), together with (27), is equivalent to (3):

Vn(ξ, t) = −M∑

m=1

qm2π∂nG(ξ, zm), ξ ∈ ∂D. (28)

The primitive of the Schwarz function (the so-calledgenerating function), Ω(z) =

∫ zS(z′), will play an im-

portant role in what follows. Let us briefly recall its mainproperties. From the series expansion of the Schwarzfunction (22) we obtain: Ω(z) = Ω+(z) + Ω−(z) − 1

2v0,where Ω±(z) are analytic in D± respectively and admitthe Taylor series expansions in the corresponding regions,

Ω+(z) =1

π

∫

D−

log(

1− z

z′

)

d2z′ =∑

k>0

tkzk,

Ω−(z) =1

π

∫

D+

log (z − z′) d2z′ = t0 log z −∑

k>0

vkkz−k.

(29)Comparing (29) with (17) we conclude that the generat-ing function can be related with the electrostatic poten-tial as follows:

Φ−(z) = −2ReΩ−(z),

Φ+(z) = 2ReΩ+(z)− v0 − |z|2.(30)

Because of the logarithm, which appears in the Taylor se-ries expansion of Ω−(z), the generating function is multi-valued while its real part if well defined. The generatingfunction is harmonic in the internal domain with a log-arithmic singularity at the origin. At the boundary’scurve, ξ ∈ Γ, the real part of Ω(ξ) equals:

ReΩ(ξ) = |ξ|2/2. (31)

The primary role of the Schwarz function in the theoryof quadrature domains [17] is also worth mentioning. Thedomain is quadrature if it is mapped to the exterior ofthe unit disk by a rational function, w(z) = P (z)/Q(z),where P (z) and Q(z) are certain polynomials. From theHerglotz theorem then follows that the Schwarz functionsof quadrature domains are meromorphic in D. Con-sider the quadrature domain D and let u(z) be an an-alytic function. Suppose that the Schwarz function ofthe boundary curve has poles of orders n1, n2, . . . , nM atthe points z1, z2 . . . , zM ∈ D respectively. By virtue of

6

the Stokes theorem and residue calculus we obtain theso-called quadrature identity:

∫

D

u(z)d2z =

∮

∂D

u(z)S(z)dz =

M∑

m=1

nm∑

j=0

cmju(j)(zm),

(32)where by u(j)(z) we denoted the j-the derivative of u(z)and the coefficients cmj ∈ C are non-zero complex con-

stants independent on u(z). The number n =∑M

m=1 nk

is called the order of the quadrature identity. Hereafterwe will refer to the poles of the Schwarz function in theexternal domain as the quadrature points.The quadrature domains appear to be important, since

any domain with a smooth boundary can be approx-imated by the quadrature one [18]. In the Laplaciangrowth governed by the point-like sources in D thequadrature domain remains to be the quadrature domainas readily follows from (24) and (32). The evolution ofthe domain can only add new poles to the Schwarz func-tion in D or change residues of the existing ones.However, as mentioned earlier, the rational solutions

of the Laplacian growth equation are known to developcusp-like singularities in a finite time. In order to de-termine singularity-free solutions, a more general classof Abelian domains [19] can be considered. These do-mains can be mapped to the w-plane by such functionw(z), that z′(w) is rational. In this case singularity-freelogarithmic solutions of the Laplacian growth problemcan be constructed [5, 6]. In this paper we will considerthe quadrature domains only, while the case of Abeliandomain will be the subject of future research.

3. Tau-function and integrability

The harmonic moments, tk and vk, are known to satisfythe symmetry relations [20]:

∂vk∂tn

=∂vn∂tk

,∂vk∂tn

=∂vn∂tk

. (33)

These relations suggest an existence of a real single-valued function of the harmonic moments tk, tk, such thatthe moments, vk and vk, can be expressed as its partialderivatives. This function, denoted as log τ , was deter-mined in [20] and identified with the logarithm of the tau-function of the dispersionless limit of the 2D Toda inte-grable hierarchy. By definition we have vk = ∂ log τ/∂tkand tk = ∂ log τ/∂vk, and similar relations hold for thecomplex conjugated moments. The logarithm of the tau-function can be written as the double integral over thedomain surrounded by the closed analytic curve [21],

log τ = − 1

π2

∫

D+

∫

D+

log

∣

∣

∣

∣

1

z− 1

z′

∣

∣

∣

∣

d2zd2z′. (34)

It can be represented as an infinite series in the harmonicmoments of the domain. Taking into account the Taylor

series expansion of the potential in the internal domain,Φ+(z), and performing the term-wise integration, we ob-tain:

log τ =1

2t0v0 +Re

∑

k>0

tkvk −1

2π

∫

D+

|z|2d2z. (35)

The integral π−1∫

D+ |z|2d2z = (1/2)t20+Re∑k>0 k tkvkcan also be expressed as a series in the harmonic mo-ments, as it readily follows from the Stokes formula.However, it is convenient to treat the last term sepa-rately.The tau-function plays an important role in studying

integrable structure of Laplacian growth. It allows oneto represent the inverse conformal map from the externaldomain to the complement of the unit disk, w(z), in thefollowing form:

logw(z) = log z − ∂t0

(

1

2∂t0 +

∑

k>0

z−k

k∂tk

)

log τ. (36)

Since the Green’s function G(z, z′) can be expressed interms of the inverse conformal map (8), one can also ob-tain its representation in terms of the tau-function,

G(z, z′) = log

∣

∣

∣

∣

1

z− 1

z′

∣

∣

∣

∣

+1

2∇(z)∇(z′) log τ, (37)

where we introduced the differential operator,

∇(z) = ∂t0 +∑

k>0

(

z−k

k∂tk +

z−k

k∂tk

)

. (38)

It is nothing but the variational derivative, acting in thespace of functionals, X = X(t0, t1, t1, . . . ), on the do-main. It can be related with the total time derivative. Byusing the chain rule and noting that the dynamics of theinterface is governed by the sources at z1, z2, . . . , zM , suchthat the harmonic moments change according to (14), wecan define the time derivative as follows:

∂

∂t=

M∑

m=1

qmπ

∇(zm). (39)

It is instructive to obtain the Laplacian growth equa-tion (26) starting from the very definition of the tau-function, provided that the time evolution of the exter-nal harmonic moments is governed by Richardson’s the-orem (14). Let us introduce the auxiliary potential,

Φ(z) = −(2/π)

∫

D+

log

∣

∣

∣

∣

1

z− 1

z′

∣

∣

∣

∣

d2z′ = ∇(z) log τ, (40)

where z ∈ D. Applying ∇(z′) to both sides of the latterrelation and expressing the double derivative of the tau-function through the Green’s function (37), we obtain:

∇(z′)Φ−(z) = 2G−(z, z′) + 2 log |w(z′)|, (41)

7

where we took into account that ∇(z)v0 = 2 log |z| −2 log |w(z)|, used the definition G−(z, z′) = G(z, z′) −log |z−z′|, and expressed the auxiliary potential, Φ(z) =Φ(z)+ v0+2t0 log |z|, in terms of the ordinary potential.Therefore from (39) and (41) it follows:

∂

∂tΦ−(z) = 2

M∑

m=1

qmπ

(

G−(z, zm) + log |w(zm)|)

. (42)

Finally, replacing the potential by the generating func-tion, Φ−(z) = −2ReΩ−(z), in accordance with (30) anddifferentiating both sides of the above relation w.r.t. z,we obtain exactly the Laplacian growth equation (26) forthe analytic in D part of the Schwarz function.

III. STOCHASTIC LAPLACIAN GROWTH

1. Random growth and classical trajectory

In this section we introduce the stochastic growth pro-cess which describes aggregation of randomly walkingparticles to form complex clusters [25]. In contrast toDLA, an arbitrary number K of uncorrelated particlesare simultaneously issued from a distant source per unittime δt. Each particle has a finite area ~, and its shapeis a curvilinear quadrangle,

√~ ×

√~, whose sides are

formed by segments of the equipotential and field lines.The initial domain is a unit circle at the origin, so that~ ≪ 1. Undergoing Brownian motion the particles reachthe growing domain and stick to its boundary, thus form-ing an external layer (with the area K~) of the advancedcluster. In the hydrodynamical limit (~ → 0) the stochas-tic source will be identified with a viscous fluid sink withthe rate q (the area growth per unit time), such that:

qδt = K~. (43)

As soon as all particles became a part of the cluster,another portion of them is released from a distant source,and the process continues like this. The case K = 1corresponds to the ordinary DLA, while the largeK limitof the model describes the stochastic Laplacian growth.The issued particles are attached to the interface with

probabilities determined by the harmonic measure [22] ofthe smooth boundary, Γi = ∂Di,

µi(ξ, z) = − 1

2π∂nGi(ξ, z)|dξ|, (44)

where i denotes the i-th time unit and Gi is a Green’sfunction of the domain Di (8). In electrostatics µi(ξ, z) isa charge distribution induced at ξ ∈ Γi by a unit chargeat z to keep the interface equipotential. In this work wewill mostly use the standard probabilistic interpretationof the harmonic measure as a probability for a Brown-ian particle issued at z to hit the boundary at the given

segment |dξ| ∈ Γi. The study of the harmonic measureis facilitated by conformal mapping since µi(ξ, z) is con-formally invariant. Because the Green’s function of thecomplement to the unit disk is

G(w,w′) = log

∣

∣

∣

∣

w − w′

1− ww′

∣

∣

∣

∣

, (45)

the harmonic measure in the w-plane reads:

µ(eiφ, w) =1

2πRe(

eiφ + w

eiφ − w

)

dφ, (46)

where dφ = |dξ|/|f ′(eiφ)| is a little arc at the unit circle.Let us divide the domain boundary uniformly into

N ≫ 1 segments with the arc-lengths |dξ| =√~, such

that adjacent particles, stuck to the interface, are at-tached to neighboring bins. The number of boundary’ssegments increase with time, as the perimeter grows.At some intermediate length scale the interface maybecome a fractal, and the number of segments scalesas N ∼ (r(t)/~1/2)dF , where r(t) is the conformal radiusof the cluster at time t and dF is the fractal dimensionof the boundary.If the number of simultaneously issued particles is

much larger than the total number of boundary segments,K ≫ N , many of them may attach to the same bin ofthe boundary. The particles are uncorrelated during asingle time step, so they all have equal attachment prob-ability to a particular bin, expressed by the harmonicmeasure (44). Since the size of each bin is

√~, the parti-

cles, which stuck to the same bin, form a column with theheight,

√~ ·k(ξn, i), where k(ξn, i) is the number of parti-

cles attached to the n-th segment of the boundary duringi-th second [42]. Let ki = k(ξ1, i), . . . k(ξN , i) be a par-ticular set of attached particles during the i-th second.Then the probability of the particles distribution alongthe boundary is given by the multinomial formula [43]:

P (ki) = K!

N∏

n=1

[µi−1(ξn,∞)]k(ξn,i)

k(ξn, i)!, (47)

where we took into account that the particles, launchedduring the i-th second, have attached to the interfaceformed during the previous (i − 1)-th second. To putit differently, the Brownian particles attached to the do-main Di−1 during i-th growth step form an external layerli = Di \Di−1 of the advanced cluster Di and (47) is thestatistical weight of this layer.As readily follows from (47), the stochastic growth is

a Markov process, i.e. the future of the growing clusteris based on its present shape. Therefore, the probabilityof the whole growth process, which results in the final

domain D(T ) = ∪T/δti=1 li, is given by the product of the

layer’s weights:

P(k) =

T/δt∏

i=1

(

K!

N∏

n=1

[µi−1(ξn,∞)]k(ξn,i)

k(ξn, i)!

)

. (48)

8

Here the symbol k = kiT/δti=1 labels different scenar-

ios (a set of distributions of attached particles duringeach growth step) of the growth process which lead tothe same final cluster. In other words, the final domainD(T ) can be divided into the layers of the same area,K~, in a number of ways; each of them represents a par-ticular growth scenario. The total statistical weight ofthe cluster, D(T ), is given by the sum of P(k) over allpossible scenarios, and can be recast in the path-integralform, which is beyond the scope of this paper. We note,however, that the standard technique of the perturbationtheory cannot be applied in this case directly, becausethere is no small parameter to distinguish different sce-narios. Below we will mainly consider the probabilities ofsingle layers (47), keeping in mind that the total proba-bility of the growth scenario is given by their product (48)over time steps.The ratio, K/N , is an important parameter of the

model. It is the average (with respect to the multino-mial distribution) number of particles attached to thesegments of the boundary per time unit. When K/N issmall, we drop a few particles onto the cluster per second.The DLA, when K = 1, can be called a quantum limitof stochastic Laplacian growth, as correlations betweenparticles in this case are maximal. The next particle al-ways “feels” a slight change of the interface, caused bya previously landed particle, while both would be totallyuncorrelated if emitted simultaneously. In the oppositelimit, K/N ≫ 1, the most probable motion of the in-terface is deterministic and obeys the Laplacian growthequation. Thus, K → ∞, is the classical limit of thistheory, which will be considered in the rest of the paperin detail.In the large K limit the model simplifies significantly.

For the large values of k(ξn, i) one can use the Stirlingapproximation for factorials, k! =

√2πk(k/e)k, to recast

the probability (47) in the form:

P (ki) = exp

−N∑

n=1

k(ξn, i) logk(ξn, i)

Kµi−1(ξn,∞)

, (49)

where the terms O(

log k(ξn, i))

were omitted in the ex-ponent. The logarithm of the probability has a form ofthe Kullback-Leibler entropy [24], which measures a dis-tance between two distributions of particles attached tothe interface, k(ξn, i) and Kµi−1(ξn,∞), generated by adistant source.Since the total number of deposited particles is K, the

numbers of attached particles, k(ξn, i), are not indepen-dent and satisfy the constraint:

N∑

n=1

k(ξn, i) = K. (50)

It is convenient to consider the continuum limit of themodel, when the number of the boundary’s segments,

N ≫ 1, is very large, while the time step, δt → 0, goesto zero. Then, the sums over discrete labels, n and i, canbe replaced by integrals over the domain’s boundary andtime in accordance to the rules:

N∑

n=1

X(ξn) =

∮

Γ

|dξ|~1/2

X(ξ),

T/δt∑

i=1

X(i) =

∫ T

0

dt

δtX(t).

(51)In the continuum limit the number of particles attachedto a particular segment of the boundary, k(ξn, i), be-comes the stochastic field [44], k(ξ, i), and (49) takes theform:

P (ki) = exp

−∮

Γi−1

k(ξ, i) logk(ξ, i)

Kµi−1(ξ,∞)

|dξ|√~

.

(52)As discussed above, the growth probabilities of sin-

gle layers (52) determine the probability of a particulargrowth scenario (48) of a certain final domain D(T ) =∏T/δt

i=1 li. However, one may ask another question: whichdomain has the maximum probability, provided that theinitial domain is a unit circle? Using a calculus of vari-ations and maximizing the functional in (52) w.r.t. thestochastic field k(ξ, i) satisfying the constraint (50) wedetermine the so-called classical trajectory:

kcl(ξ, i) = Kµi−1(ξ,∞), (53)

The second variation of the functional at kcl(ξ, i) isstrictly negative and, therefore, the classical trajectoryprovides a global maximum of the probability. It be-comes exponentially sharp in the classical limit, ~ → 0,such that fluctuations around the saddle point vanish.To determine the probability of the classical trajectory,the previously omitted factor,

√2πk, of the Stirling ap-

proximation should be taken into account [45]. Then, weobtain:

P (kcli+1) = exp

−1

2

∮

Γi

log(K√~ |w′

i(ξ)|)|dξ|√~

, (54)

where we expressed the harmonic measure in terms of theGreen’s function (44) and took into account that |dξ| =√~ and ∂nG(ξ,∞) = −|w′(ξ)| for ξ ∈ Γ.Remarkably, the interface dynamics corresponding to

the classical trajectory (53) is governed by the determin-istic Laplacian growth equation. Indeed, the normal dis-placement of the boundary, generated by attached par-ticles, is δh(ξ, i) =

√~ · k(ξ, i). Introducing the normal

boundary velocity as Vn(ξ, i) = δh(ξ, i)/δt, we determinethe interface dynamics, corresponding to the classical tra-jectory:

V cln (ξ, i) = − q

2π∂nGi−1(ξ,∞), (55)

where we used (43) to express the number of particlesthrough the rate of the source. Thus, the normal bound-ary velocity is proportional to the pressure gradient, as it

9

should be according to the Darcy’s law. The conventionalLaplacian growth equation can be derived by means ofthe kinematical identity for the normal velocity of theboundary (9). Therefore, for the classical trajectory weobtain the deterministic Laplacian growth equation forevolution of the interface governed by single hydrody-namical source at infinity:

Im(

∂tf(φ, t)∂φf(φ, t))

=q

2π. (56)

Thus, it turned out possible to derive the Lapla-cian growth equation directly from a variational calculusbased on elementary combinatorics. So far this equationwas possible to deduce only from viscous hydrodynamicsor kinetics [23].The LG equation (56) describes the uniform growth

of the initially unit disk. However, experiments in theHele-Shaw cell are known to produce different complexuniversal patterns reflecting highly unstable and non-equilibrium nature of the Laplacian growth. In the nextsubsection we will address growth of these complex pat-terns from the stochastic point of view. One may stillwonder why complex clusters emerge in experiments, be-cause their probabilities are exponentially suppressed (inthe limit ~ → 0) as compared to the classical one. Thereason is the exponentially large numbers of growth sce-narios of complex shapes, which makes them possibleeven in the classical limit.

2. Non-classical trajectories and virtual sources

In this section we consider non-classical scenarios of thestochastic Laplacian growth. These scenarios are char-acterized by a non-uniform attachment of Brownian par-ticles to the interface at a time and this will result intocomplex shapes of the final clusters. As discussed above,any planar domain with smooth boundary can be approx-imated by the quadrature domain as closely as desired.It will be shown that the quadrature points of the cor-responding domain can be treated as the hydrodynamicsources, which govern the evolution of the interface. Fi-nally, we will express the statistical weights of complexdomains in terms of their quadrature points.Any distribution, k(ξ, i), of particles, attached dur-

ing i-th growth step, specifies uniquely the deformationof the cluster and therefore can be related to the cor-responding variation of the Schwarz function. For anyquadrature domain the most general variation of theSchwarz function (more precisely S+(z)) can be writtenas a sum of simple poles:

S+i (z) = S+

i−1(z)−1

π

M(i)∑

m=1

ǫm(i)

z − zm(i), (57)

where S+i−1(z) and S+

i (z) are the Schwarz functions atthe beginning and at the end of the i-th growth step re-

spectively. The higher order poles, which could appearin the r.h.s. of this expression, can be treated as a cer-tain combinations of simple ones, which almost coincide.Since k(ξ, i) change with time randomly in the stochasticgrowth process, so do the poles of ∆S+

i (z). For brevity,we will omit time dependence of M(i), zm(i) and ǫm(i),unless it brings confusion.

Let us explicitly relate the distribution of attachedparticles along the boundary, k(ξ, i), with the poles of∆S+

i (z). For this purpose consider a set of M hydrody-namical sources at the points zm in the external domain.Suppose that the rate of each source, qm, is given by

qm = ǫm/δt. (58)

As discussed in the Introduction, the interface dynam-ics, governed by these sources, and the Darcy’s law aredescribed by the Laplacian growth equation (26). In par-ticular, the time derivative of S+

i (z) is given by the sumof simple poles (24). Therefore, the increment of theSchwarz function due to the aggregation process (57),can be treated as the result of the deterministic Lapla-cian growth, governed by the set of virtual (fictitious)hydrodynamical sources. Their positions and rates areuniquely specified by the poles of ∆S+

i (z).

The two-dimensional hydrodynamic (electrostatic) po-tential problem has the following essential feature: eachsource (a positive charge) located at zm in the com-plex plane, has with necessity the corresponding sink(a negative charge) of the same rate at infinity. Itcan be easily seen from the two-dimensional potential,Φ(z) = qm log |z − zm|, created by the source qm at zm.The potential has two logarithmic branch points, z = zmand z = ∞, which correspond to the locations of charges.Therefore, in addition to virtual sources, correspondingto the poles of ∆S+

i (z), one needs to take into accountthe additional set of sinks at infinity. In other words, in-stead of virtual sources one can introduce virtual dipoles :a separation by large distance of sources (at zm) andsinks (at ∞) with equal rates. Because of these addi-tional sinks at infinity, the virtual dipoles change onlythe external harmonic moments of the domain, tk and tk(k > 0), but not its area, t0, as readily follows from (14).The area growth of the cluster is governed by the actualsource at infinity, which deposits K particles at a time.It will be convenient to consider the sources at infinityaltogether as a single, say, M -th, source.

Hereafter we will refer to simple poles of ∆S+i (z) as to

the active virtual sources at the i-th growth step. Thevirtual sources, which were active in the past (i.e. thepoles of ∆S+

j (z) with j < i), will be called the frozen

sources w.r.t. the i-th second. Note that the sources canbe active during finite time intervals. In this case, nonew poles of Schwarz function appear, only the residuesof the existing poles change. The final pattern we arrivedto at the end of the growth process is the quadrature

10

domain, and its degree equals to the total number offrozen sources, i.e. all poles of S+(z, T ) at time T .

3. Stochastic growth with several sources

So far the stochastic growth process was generated bya single source at infinity, and the most probable inter-face dynamics was described by the LG equation (56).In other words, the stochastic source mimics the hydro-dynamic one in the classical regime. Now we will con-sider the stochastic growth process generated by severalsources of Brownian particles.Let M uncorrelated sources located at z1, . . . , zM ∈ D

deposit K1, . . . ,KM particles respectively onto the grow-ing cluster during the i-th growth step. Since the totalnumber of particles attached to the boundary at the timeunit is K, the following constraint holds:

K1 +K2 + · · ·+KM = K. (59)

Let km(ξn, i) be a number of particles launched fromthe m-th source and attached to the n-th segmentof the boundary during the i-th second. By ki =km(ξ1, i), . . . , km(ξN , i)Mm=1 we denote the attachedparticles, which form the layer li. Since all particles areuncorrelated within a growth step the probability of thelayer is given by the product of M multinomial distribu-tions, each representing the corresponding source:

P (ki) =M∏

m=1

N∏

n=1

[µi−1(ξn, zm)]km(ξn,i)

km(ξn, i)!, (60)

By virtue of the Stirling approximation and replacingthe sum of boundary segments in the exponent by theintegral along the boundary (51), we recast (60) in theform [46]:

P (ki) = exp

−M∑

m=1

∮

Γ(i−1)

km(ξ, i) ×

× logkm(ξ, i)

Kmµi−1(ξ, zm)

|dξ|√~

. (61)

Similar to (49), each term in the logarithm of (61) isa Kullback- Leibler entropy, which measures a distancebetween distributions, km(ξ, i) and Kmµi−1(ξ, zm), gen-erated by the source at zm.One may ask which final domain is the most probable

in presence of several sources? By applying a calculus ofvariations and maximizing (61) w.r.t. stochastic fieldskm(ξ, z) subjected to the constraint (59), one obtainsM independent transcendental equations for the saddlepoint, which have the solutions:

kclm(ξ, i) = Kmµi−1(ξ, zm). (62)

Being attached to the interface the particles force thedomain to grow, and the partial contribution of the m-thsource to the normal interface velocity reads:

V (m)n (ξ, i) = −qm

2π∂nGi(ξ, zm), (63)

where qm = Km~/δt is a rate of the m-th hydrodynam-ical source. The total boundary velocity is given by the

sum of single contributions: Vn(ξ, i) =∑M

m=1 V(m)n (ξ, i).

The additional sources cause a deflection of the ini-tially unit disk from its circular shape during the growthprocess. Variation of the harmonic measure along theboundary results in the non-uniform growth: the fastergrowing tips shield the other parts of the cluster, as thefjords (empty spaces between growing fingers) becomeless accessible to incoming particles. Expressing the nor-mal velocity of the boundary in terms of the conformalmap (9), we obtain the Laplacian growth equation inpresence of M hydrodynamical sources (11). Thus, thestochastic sources of Brownian particles mimic the hy-drodynamic ones in the classical limit. We will use theproposed interpretation of hydrodynamical sources to ex-press the statistical weights of non-classical growth sce-narios in terms of the quadrature points of the growingdomain.

4. Statistical weights of non-classical scenarios

Consider stochastic growth of a quadrature domain,such that ∆S+

i (z) during the i-th time step is givenby the sum of simple poles (57). What is the proba-bility of the generated layer li = Di \ Di−1? As dis-cussed above, the poles of ∆S+

i (z) can be treated asthe (virtual) hydrodynamical sources, or as the (virtual)stochastic source of Brownian particles, operating in theclassical regime. However, since the Brownian particlesare actually launched from the distant source at infin-ity, the probability of any distribution of attached par-ticles is determined by the multinomial formula for theinfinitely remote source (49). The weight of the layerkcli = kclm(ξ1, i), . . . , k

clm(ξN , i)Mm=1, which is described

by the distribution (62) of attached particles and cor-responds to the variation (57) of the Schwarz function,reads:

P (kcli ) =

M∏

m=1

exp

−N∑

n=1

kclm(ξn, i) log

(

kclm(ξn, i)

Kµi−1(ξ,∞)

)

.

(64)Here we took into account that the distributions of par-ticles, kclm(ξn, i), launched from the different sources, areuncorrelated. Finally, in the continuum limit (N → ∞)the probability of a single layer takes the form:

P (kcli+1) = Ni+1 exp

−M∑

m=1

Km

∮

Γi

µi(ξ, zm) ×

11

× log

∣

∣

∣

∣

∂nGi(ξ, zm)

∂nGi(ξ,∞)

∣

∣

∣

∣

, (65)

where we used (62) to express kclm(ξ, i) in terms of har-monic measure. By Ni we denoted the pre-factor, whichemerges after taking Km/K (recall that kclm ∼ Km) outof logarithm in (64),

Ni = exp

K logK −M∑

m=1

Km logKm

. (66)

It is worth mentioning, that Ni has a clear statisticalinterpretation. Suppose that K randomly walking parti-cles, launched from the distant source, are indistinguish-able. There is a number of ways to sort them into Mgroups; each group represents the corresponding virtualsources and contains Km particles. More precisely, thereare exactly CK

K1ways to choose K1 particles out of K,

CK−K1

K2ways to chose K2 particles out of K−K1, and so

forth, where by Cnk we denoted the binomial coefficients.

By multiplying the binomial coefficient and simplifyingthe result, we obtain the combinatorial factor:

Ni =K!

K1(i)!K2(i)! · · ·KM(i)(i)!, (67)

which becomes (66) after applying the Stirling approxi-mation. The combinatorial pre-factor can be estimatedas follows. Let the number of active sources, M , duringeach growth step be large and constant in time. Thenumber of building blocks, Km, deposited from eachsource, can be roughly estimated as Km ∼ K/M . Thus,in the leading order in M we obtain that Ni ∼MK

i .

5. Growth probabilities and quadrature points

In the exponent of (65) one can recognize the Dirichletformula for harmonic continuation of a certain bound-ary function to the point zm in the external domain D.Therefore, it seems reasonable to recall briefly some ele-ments of the problem. Without loss of generality considera simply-connected domain D with a piecewise-analyticboundary. The Dirichlet problem [12] is to find a har-monic function in the exterior of the domain, such thatit is continuous up to the boundary and equals a givenfunction u0(z) on the boundary. We will refer to thisfunction as to the harmonic extension of u0(z) to theexterior and will denote it by uH(z). It is given by thePoisson integral formula (68):

uH(z) = − 1

2π

∮

∂D

∂nG(ξ, z)u0(ξ)|dξ| ≡∮

Γ

µ(ξ, z)u0(ξ),

(68)Now we are ready to determine the harmonic exten-

sions of the required functions in (65) explicitly. Let usfirst consider the numerator of the logarithm. Since the

contributions of different sources factorize, we considerthem independently. By virtue of the Laplacian growthequation for the Schwarz function (26) together with (39)we obtain |∂nG(ξ, zm)| = |∇(zm)S(ξ)|. From (24) it fol-lows that ∇(zm)S+(ξ) = −1/(ξ − zm). Therefore, theharmonic extension of the logarithm of the normal deriva-tive of the Green’s function can be recast in the form:

∮

Γ

µ(ξ, zm) log |∂nG(ξ, zm)| =

=

∮

Γ

µ(ξ, zm) log

∣

∣

∣

∣

1

ξ − zm−∇(zm)S−(ξ)

∣

∣

∣

∣

, (69)

where we used the decomposition of the Schwarz functionS(ξ) = S+(ξ) + S−(ξ) introduced earlier (22). Reducingthe expression in the logarithm to a common denomina-tor, we rewrite the previous formula as follows:

∮

Γ

µ(ξ, zm) log |∂nG(ξ, zm)| =∮

Γ

µ(ξ, zm)×

×(

log∣

∣1− (ξ − zm)∇(zm)S−(ξ)∣

∣− log |ξ − zm|)

.

(70)

It is useful to recall a harmonic extension of the logarithmfrom the boundary to a certain point at the bulk:∮

Γ

µ(ξ, z) log |ξ−zm| = log |z−zm|−G(z, zm)+G(z,∞),

(71)where the Green’s functions in the r.h.s do not changethe boundary value of u0(ξ) = log |ξ − zm| and makeit harmonic in the whole domain, including the pointsz = zm and z = ∞. By means of (71) we determine therequired harmonic extension of the last term in the r.h.s.of (70):

∮

Γ

µ(ξ, zm) log |ξ − zm| = −G−(zm, zm) +G(zm,∞),

(72)where the function G−(z, z′) = G(z, z′) − log |z − z′| isharmonic everywhere in D, except infinity.Consider the former logarithm in the r.h.s. of (70).

From the Taylor series expansion of S−(z) in D (22), weobtain:

∇(zm)S−(z) =1

z+∑

k>0

∇(zm)vk z−k−1, z ∈ D. (73)

Therefore, log |1− (ξ − zm)∇(zm)S−(ξ)| is a harmonicfunction of ξ everywhere in the external domain, ex-cept infinity, where is diverges logarithmically, − log |ξ|.To make this function harmonic at infinity we add theGreen’s function, −G(ξ,∞), which does not change theboundary’s value. Note that at the point, z = zm, thecumbersome contribution of (z − zm)∇(zm)S−(z) underthe logarithm vanishes and the value of the harmonicallycontinued function at the point z = zm is represented by

12

the Green’s function only: −G(zm,∞). Thus, combiningthe contributions of both logarithms from (70), we arriveto the following identity:

∮

Γ

µ(ξ, z) log |∂nG(ξ, z)| = G−(z, z)− 2G(z,∞). (74)

Finally let us consider the denominator of the log-arithm in (65). The normal derivative of the Green’sfunction reduces to ∂nG(ξ,∞) = −|w′(ξ)| at the bound-ary. The inverse conformal map, w(z), admits the fol-lowing Laurent series expansion in the external domain:w(z) = z/r+

∑

j≥0 pjz−j , where the coefficient at leading

term, r (the conformal radius of the domain), is chosen tobe real and positive. Since the function log |∂nG(ξ,∞)| =log |w′(ξ)| is harmonic everywhere in the external domain(including infinity) and continuous up to the boundary,its harmonic extension reads:

∮

Γ

µ(ξ, zm) log |∂nG(ξ,∞)| = log |w′(zm)|. (75)

Summarizing, the probability (65) can be written interms of the quadrature points of the domain, which wereactive sources during the growth step:

P (kcli+1) = Ni+1 exp

−M∑

m=1

Km

(

G−i (zm, zm)−

− 2Gi(zm,∞)− log |w′i(zm)|

)

. (76)

This expression is not the final way to present the re-sult. There are two natural ways to proceed: to rewriteP (kcl

i ) in the w-plane, expecting to get some simplifica-tion, and/or to recast the statistical weight in a certainfunctional on the domain in z-plane. As we will showboth ways are fruitful. Let us address them in succes-sion.

6. Growth probability in the w-plane

The transformation of the growth probability to thew-plane is facilitated by the conformal invariance of theharmonic measure. Since ∂nGi(ξ,∞) = −|w′

i(ξ)|, theratio under the logarithm in (65) transforms to

∂nGi(ξ, zm)

|w′i(ξ)|

= ∂nG(eiφ, wm), (77)

where wm = wi(zm) is the pre-image of the source in theexterior of the unit disk. By G(w,w′) with arguments inthe w-plane we will denote the Green’s function of thecomplement to the unit disk. Therefore, we obtain:

P (kcli+1) = Ni+1 exp

−M∑

m=1

Km

∮

|w|=1

µ(w,wm) ×

× log |∂nG(w,wm)|

. (78)

Since ∂nG(eiφ, wm) = −(|wm|2− 1)/|eiφ−wm|2, and the

harmonic measure is µ(eiφ, wm) = −∂nG(eiφ, wm)dφ/2π,the residue calculus allows to evaluate the contour inte-gral in the exponent of (78) without any references tothe Dirichlet boundary problem:

∮

|w|=1

µ(w,wm) log |∂nG(w,wm)| = − log(1− |am|2),

(79)where am = 1/wi(zm) are time-dependent inverted pre-images of active sources inside the unit circle. Since thepoints zm are singularities of the Schwarz function inD, the Herglotz theorem identifies the points am withsingularities of the conformal map z = f(w) inside theunit disk.

With help of (79) the probability (78) for a single layercan be reduced to the following neat expression:

P (kcli+1) = Ni+1 exp

M∑

m=1

Km log(

1− |am|2)

. (80)

In the limit δt→ 0, the probability of a particular growthscenario (48) of the final domain D(T ) becomes

P(kcl) =

T/δt∏

i=1

Ni

×

× exp

1

~

∫ T

0

M∑

m=1

qm(t) log(1 − |am(t)|2) dt

, (81)

where the rate qm(t) = Km(t)~/δt is a characteristicfunction of the source. When qm(t) is a delta-function,the source operates during a single growth step only. Forsmooth characteristic functions, qm(t), the integrals ofmotion zm = f(1/am) provide time-dependence of am.Remarkably, each term in the exponent of (80) has a

clear electrostatic interpretation. It is an electrostatic en-ergy of interaction between a unit charge at am and thecorresponding induced charges on the (grounded) unitcircle, to keep zero potential on it. The absence of inter-action between the charges at am themselves is a manifes-tation of the sources’ independence during each growthstep.To justify an electrostatic interpretation of the loga-

rithm of the growth probability, we recast the contourintegral (79) in the double contour integral. Since in thew-plane ∂nG(e

iφ, wm) = (|wm|2 − 1)/|eiφ − wm|2, and∮

|w|=1 µ(w,wm) = 1, we obtain:

∮

|w|=1

µ(w,wm) log|wm|2 − 1

|w − wm|2 = log(|wm|2 − 1)−

13

− 2

∮

|w|=1

µ(w,wm) log |w − wm|. (82)

The latter logarithm, log |w − wm|, in the r.h.s. can berepresented as a solution of the Dirichlet boundary prob-lem. Since |w| = 1 we obtain:

log |w−wm| =∮

|w′|=1

µ(w′, wm) log |w−w′|−G(wm,∞),

(83)where G(wm,∞) = − log |wm|. Combining (82) and (83)we recast a single contour integral (79) in the doublecontour one:

∮

|w|=1

µ(ξ, zm) log |∂nG(ξ, wm)| =

= −∮

|w|=1

∮

|w′|=1

µ(w,wm) log |w − w′|µ(w′, wm),

(84)

which is an electrostatic energy of self-interacting chargesinduced on the unit circle with density µ(ξ, wm). Again,due to sources independence the terms, which describethe interaction between charges at the points wm, areabsent in the final expression. This is a reason (as wewill show) why the probability of the layer can not bewritten as a single functional on the layer, but rather asa sum of independent functionals.Therefore, the Kullback-Leibler entropy in the expo-

nent of (64) was transformed to electrostatic energy and,therefore, the probability of a single layer (80) takesa form of the Gibbs-Boltzmann distribution, which be-comes more obvious with a help of (84):

P (kcli+1) = Ni+1×

×exp

M∑

m=1

Km

∮ ∮

µ(w,wm) log |w − w′|µ(w′, wm)

,

(85)

where the integrations go along unit circles. Strikingly,despite non-equilibrium nature of LG, the growth proba-bilities have the Gibbs-Boltzmann form, which tradition-ally is not applied out of equilibria.

7. Growth probability in the z-plane

In the z-plane one can transform the growth probabil-ity in a certain functional on the layer, which consistsof all particles, attached to the interface during a singlegrowth step. Contribution from the numerator of thelogarithm (65) can be transformed in the double contourintegral along the interface, similar to (84). By using

∮

Γ

µ(ξ, zm) log |∂nG(ξ, zm)| =

= −∮

Γ

µ(ξ, zm) log |ξ − zm| −G(zm,∞), (86)

which is another representation of (71), and

log |ξ − zm| =∮

Γ

µ(ξ′, zm) log |ξ − ξ′| −G(zm,∞), (87)

we arrive to the identity:

∮

Γ

µ(ξ, zm) log |∂nG(ξ, zm)| =

= −∮

Γ

∮

Γ

µ(ξ, zm) log |ξ − ξ′|µ(ξ′, zm). (88)

The Stokes formula allows one to transform the dou-ble line integrals along the domain’s boundary to a cer-tain functional on the layer itself. However, since theintegrand in (88) is a logarithm of the distance betweenboundary points, one needs to pay an extra attention tologarithmic cuts inside the layer. As follows from theLaplacian growth equation (26), the normal derivativeof the Green’s function can be written in terms of theSchwarz function. Projecting both side of the Lapalciangrowth equation to the unit normal nξ = 1/

√

−S′(ξ)and taking into account that the contribution of differentsources factorizes during a single time step, we obtain:

∂nGi(ξ, zm) = −nξ · ∇(zm)Si(ξ), (89)

where the differential operator ∇(z) was introducedin (38) and related to the time derivative in (39). Bymeans of (89) and noting that inξ · |dξ| = dξ we recastthe double line integral (88) in the form:

∮

Γ

µ(ξ, zm) log |∂nG(ξ, zm)| = −Re∮

Γ

∮

Γ

dξ

2πi

dξ′

2πi×

× log(ξ − ξ′)∇(zm)S(ξ)∇(zm)S(ξ′), (90)

Let us denote by l(m)i the layer, which consist only of

Brownian particles, which were deposited from the m-thactive source and attached to the domain Di−1. By def-

inition, the layers l(1)i , l

(2)i , . . . have the common internal

boundary Γi−1 = ∂Di−1, while the external boundariesdiffer. Thus, these layers are not the physical layer of the

domain, i.e.∑

m l(m)i 6= li, except the case, when a single

source operates during a growth step.For notation simplicity to the end of this section it

will be assumed that each layer is generated by a sin-gle source. The time intervals will be labeled accordingto the source, which operates during this intervals, i.e.the m-th source, operating during the m-th growth step,generates the layer lm, i.e. in (90) we set i = m. Underthese assumptions, we can replace the ∇(z) operators bythe ratio of infinitesimal differences:

∇(zm)Sm−1(ξ) = (π/qmδt)(Sm(ξ)− Sm−1(ξ)), (91)

14

where by Sm−1(ξ) and Sm(ξ) we denoted the Schwarzfunctions of the boundary curves in the beginning and inthe end of the m-th growth step correspondingly.It is convenient to introduce an auxiliary integral:

I(z) =Km~

π

∮

Γm−1

µ(ξ, zm) log |z − ξ| (92)

Using the definition of the harmonic measure (44), to-gether with (89) and (91), we can transform I(z) as fol-lows:

I(z) = Re∮

Γm−1

(

Sm(ξ)−Sm−1(ξ))

log(z− ξ) dξ2πi

. (93)

Let us move the contour, Γm−1 → Γm, in the first in-tegral in (93). Since the point z is inside of the layer,one needs to pay special attention to the logarithmic cutwhich starts in the point z and intersect Γm at somepoint, say ξ0 for definiteness. Then, the required integraltakes the form:

∮

Γm−1

Sm(ξ) log |z − ξ| dξ2πi

= −Re∫

[z,ξ0]

Sm(ξ)dξ+

+

∫

Γm\ξ0

Sm(ξ) log |z − ξ| dξ2π, (94)

where the former integral in the r.h.s. goes along the bothedges of the cut [z, ξ0] in a clockwise direction. Now, letus add and subtract the following difference of two lineintegrals (along the lower, [z0, ξ], and the upper, [ξ, z0],edges of the cut) to I(z):

Icut(z, ξ0) = Re(

∫

[ξ0,z]

−∫

[z,ξ0]

)

Scut(ξ)×

× log(z − ξ)dξ

2πi= −Re

∫

[ξ,z0]

Scut(ξ)dξ, (95)

where by Scut(ξ) we denoted the Schwarz function of thecut. One can apply the Stokes theorem to recast the sumof the following integrals along the layer’s boundaries,IΓm\ξ0 + Icut − IΓm−1

, in the integral over the layer lm.Thus, we obtain:

I(z) =1

π

∫

lm

log |z − z′|d2z′−

−Re∫

[z,ξ0]

Sm(ξ)dξ +Re∫

[z,ξ0]

Scut(ξ)dξ. (96)

The latter two integrals can be expressed in terms of thegenerating functions, Ωm(z) =

∫ zSm(z′), of the contour

Γm and the cut [z, ξ0] correspondingly. SinceReΩm(z) =|z|2/2 for z ∈ Γm, we transform I(z) in the form:

I(z) =1

π

∫

lm

log |z − z′|d2z′ − |z|22

+ReΩm(z). (97)

If the point z ∈ Γm−1 belongs to the inner boundaryof the layer, one can replace |z|2/2 in (97) by the realpart of the generating function, ReΩm−1(z). By virtueof (96) one can recast the double contour integral (90) inthe double integral over the layer lm = Dm \Dm−1:

− ǫ2m

∮

Γm−1

µ(ξ, zm) log |∂nG(ξ, zm)| =∫

lm

∫

lm

log |z − z′| d2zd2z′ − π

∫

lm

Am(z)d2z, (98)

where by ǫm = Km~ we denoted an area of the layer, andAm(z) is the modified Schwarz potential [30]:

Am(z) =|z|22

−ReΩm(z). (99)

Since ReΩm(z) = |z|2/2 for z ∈ Γm, the modifiedSchwarz potential vanishes at the boundary’s curve. Be-sides, since its first derivatives vanish for all points ofthe boundary, ∂Am(z) = ∂Am(z) = 0 for z ∈ Γm, thecontour z = Sm(z) is its saddle point. The derivationof (98) is straightforward. The only subtlety is the fol-lowing identity:

∫

lmReΩm−1(z) =

∫

lmReΩm(z).

As for the denominator of the logarithm in (65) it wasshown to be equal to (75). Thus, using (98) we recast theprobability of the layer (76), generated by a singe sourceat the point zm, in the form:

P (kclm) = Nm|w′

m−1(zm)|Km exp

1

~2Km×

×(∫

lm

∫

lm

log |z − z′| d2zd2z′ − π

∫

lm

Am(z)d2z

)

.

(100)

An explicit dependence of the growth probability (100)(which should be a functional on the layer) on thequadrature point, zm, should not be confusing, sincelog |w′(zm)| =

∫

lmlog |w′(z)|d2z.

Interestingly, the growth probability (100) in the z-plane can be also obtained within the Dyson gas ap-proach, which treats the internal domain, D+, as an elec-tronic droplet in the non-uniform magnetic field. Thesemi-classical dynamics of the droplet, which is gener-ated by adding of electrons to the boundary, is relatedto the Laplacian growth problem [31]. In particular, thesemi-classical probability to add a single electron to thepoint z0 ∈ Γ is given by the square amplitude of theone-particle wave function [31]:

|ψ(z0)|2 =|w′(z0)|√

2π3~exp

− 2

~A(z0)

, (101)

where A(z) is the modified Schwarz potential of theboundary of the droplet.If only a single Brownian particle is attached to the

interface, Km = 1, during the growth step, the layer,

15

lm, is reduced to a single point, say z0, at the boundary.Then, the double integral in (100) along the layer disap-pear [47], and using K log |w′(zm)| =

∫

lmlog |w′(z)|d2z

we obtain:

P (kclm) ∼ |w′

m−1(z0)| exp

−π~Am−1(z0)

, (102)

which coincides with (101) up to an overall normalizationconstant and the normalization factor (redefinition of ~)in front of A(z).Therefore, it is reasonable to propose that the layers’

probability (100) corresponds to the square modulus ofthe many-particle wave-function, |ψ(z1, . . . , zN )|2, suchthat the points z1, . . . , zN ∈ lm, in the Dyson gas ap-proach. In particular, an exponent of the double integralover the layer,

∫∫

log |z−z′|, in (100) is the Vandermondedeterminant of the many-particle (Slater-Jastrow) wavefunction.The Dyson gas representation of the growth probabil-

ity (100) allows one to relate stochastic Laplacian growthwith the theory of normal random matrices [32, 33] (seealso [34] for large N expansion). In [35] the growth (de-fined as increasing the total number of eigenvalue whilekeeping the potential fixed) of normal random matrixensemble was studied. From the stochastic Laplaciangrowth point of view, this is the deterministic growthprocess, governed by a single source at infinity. In orderto obtain (100), an increasing of total number of eigen-values should be supplemented with a variation in timeof a potential for the random matrices.

8. Scenario-independent part of probability

In the previous section we transformed the growthprobability of the layer in a certain functional on thelayer. According to (48) the probability of a particu-lar scenario to arrive to the final domain is given by theproduct of the layers’ probabilities. It can be shown, thatprobabilities of distinct scenarios are not equal. However,contributions of factors |w′

i−1(zm)|Km are scenario inde-pendent, and

MD =

T/δt∏

i=1

exp

M∑

m=1

Km log |w′i−1(zm)|

. (103)

depends on the final domain only.The logarithm, log |w′(zm)|, is a harmonic function in

D and can be written as the solution to the Dirichletboundary problem (68):

log |w′(zm)| =∮

Γ

µ(ξ, zm) log |w′(ξ)|. (104)

Since log |w′(z)| is regular inside the layer li, by virtue of(89) and (91), we transform (104) as follows:

M∑

m=1

Km log |w′i−1(zm)| = 1

~Re(∮

Γi

Si(ξ)−

−∮

Γi−1

Si−1(ξ)

)

log(w′i−1(ξ))

dξ

2√−1

, (105)

By changing the summation label, i − 1 → i, in the lastcontour integral, and using (51) to rewrite the discretesummation over time steps in terms of the continuousintegral over time, we recast MD in the form:

~ logMD =

∫ T

0

dt

δtRe∮

Γt

St(ξ) log

(

w′t−δt(ξ)

w′t(ξ)

)

dξ

2i+

+

(∮

ΓT

ST (ξ) log |w′T (ξ)| −

∮

Γ0

S0(ξ) log |w′0(ξ)|

)

dξ

2i.

(106)

The last term in this expression disappears if the ini-tial domain is a unit disk. Let us replace w′(z) by1/f ′(w) and expand the ratio of two conformal maps atneighbor time intervals in δt up to the first order terms:ft−δt = ft − ftδt. Using dξ = f ′(eiφ)dφ and expandingthe logarithm in small δt we transform the first term inthe r.h.s. of (106) to

1

2~

∫ T

0

dt Im∫ 2π

0

ft(eiφ)f′t(e

iφ)dφ =Area

~−

− 1

2~

∫ T

0

dt

∫ 2π

0

Im(

∂tft(eiφ)f′t(e

iφ))

dφ. (107)

The last expression was obtained through integration byparts, and by Area we denoted an area of the growndomain, DT \ D0. Using the Laplacian growth equa-tion (11), we replace the integrand by the contribution of

the corresponding sources. Since∫ 2π

0Re(eiφ + a)/(eiφ −

a)dφ = 1, the integral over φ in (107) depends only onthe number of Brownian particles, deposited from eachsource during each time step. Denoting the total numberof the particles by Area/~ we arrive to the following neatexpression:

MD = exp

Area

2~+

1

~

∮

ΓT

ST (ξ) log |w′T (ξ)|

dξ

2i

.

(108)Thus, the probability of a particular growth scenario of

the domain D can be written as the following functionalon the layers of the domain (each layer is generated bya single source, according to the analysis of the previoussection):

P(k) = NMD

T/δt∏

m=1

exp

1

~2Km×

×(∫

lm

∫

lm

log |z − z′| d2zd2z′ − π

∫

lm

Am(z)d2z

)

.

(109)

16

9. Free energy of the layer and Schwarz potential

The exponent of (109) can be rewritten through theharmonic moments of the layer, tlk and vlk, which aredefined similarly to those of planar domains (13),

tlk = − 1

πk

∫

l

z−kd2z, tl0 =1

π

∫

l

d2z,

vlk =1

π

∫

l

zkd2z, vl0 =2

π

∫

l

log |z|d2z,(110)

where the layer l is an integration region. The externalharmonic moment, tl0, coincides with the area (dividedby π) of the layer, while vl0 is the (minus) electrostaticpotential, created by the layer at the origin. The mo-ments of the layer li which is generated by M sources,can be written as the solutions to the Dirichlet boundaryproblem:

tlik =M∑

m=1

ǫmπk

∮

Γi−1

µ(ξ, zm) ξ−k,

vlik =

M∑

m=1

ǫmπ

∮

Γi−1

µ(ξ, zm) ξk,

(111)

where ǫm = Km~ is an area of particles, deposited fromthe m-th source per time unit. Similar to the simply-connected domain, the external moments tlk do not de-pend on the layer itself, but only on the position and rateof the generating source (quadrature point), while vlk alsodepend on the internal boundary of the layer:

tlik =M∑

m=1

ǫmπ

z−km

k, vli0 = −2

M∑

m=1

ǫmπ

log

∣

∣

∣

∣

w(zm)

zm

∣

∣

∣

∣

,

tli0 =

M∑

m=1

ǫmπ, vlik =

M∑

m=1

ǫmπ

(

zkm − ∂tk ReΩi−1(zm))

,

(112)Thus, one can relate the internal harmonic moments ofthe layer with the higher Hamiltonians,Hk(z) = ∂tkΩ(z),introduced in [21].The set of external harmonic moments of the layer, tlk,

is not enough to restore the internal moments, vlk, andadditional data is required, e.g. the external harmonicmoments of the interior boundary of the layer.The growth probability of the layer (100) can easily be

rewritten through the following integral

fi = − 1

π2

∫

li

∫

li

log

∣

∣

∣

∣

1

z− 1

z′

∣

∣

∣

∣

d2zd2z′ =

= − 1

π2

∫

li

∫

li

log |z − z′|d2zd2z′ + tli0 vli0 , (113)

which coincides with the tau-function for analyticcurves (34) after replacing the layer li by a simply con-nected domain D. Electrostatic interpretation of the in-tegral fi is straightforward. Suppose the layer li is filled

homogeneously with an electric charge of unit density.Then, the integral fi is equal (up to the term tli0 v

li0 ) to

the electrostatic energy of the layer. In what follows, wewill refer to (113) as to the free energy of the layer. It isa real function of the external harmonic moments of thelayer and its internal boundary and can be expanded inthe infinite series:

fi =1

2tli0 v

li0 +Re

∑

k>0

tlik vlik − 1

π

∫

li

Ai−1(z)d2z, (114)

where Ai is the modified Schwarz potential (99). Thedifference between (114) and the similar series represen-tation of the tau-function of the boundary curve (35) isan additional contribution of the generating functional,Ωi−1(z), of the internal boundary of the layer. In partic-ular, if the layer coincides with the whole domain, i.e. itsinternal boundary is absent, the free energy (113) coin-cides with the tau-function of the domain.The free energy, fm, of the layer, lm = Dm \ Dm−1,

generated by a single source at the point zm, is related tothe tau-function of the boundary curve, Γm−1, as follows:

1

2(tlm0 )2∇(zm)∇(zm) log τm−1 = fm+

+1

π

∫

lm

Am−1(z)d2z =

1

2tlm0 vlm0 +Re

∑

k>0

tlmk vlmk ,

(115)

where we denoted by log τm the logarithm of the tau-function (34), associated with the domainD+ = Dm, andused (114) in the last equality. Therefore, the probabilityof the layer (100), can be written in terms of the harmonicmoments of the layer:

P (km) = Nm|w′m−1(zm)|Km exp

π2

~2Km×

×(

1

2tlm0 vlm0 −Re

∑

k>0

tlmk vlmk

)

. (116)

Remarkably, P (km+1) can be recast in the neat andcompact expression in terms of the tau-function (34) ofinternal boundary of the layer Γm. Rewriting the con-formal map, w(z), and the Green’s function through thetau-function of the domain, (36) and (37), the followingidentity can be proven [21]:

− log

(

1− 1

|w(zm)|2)

=

=

(

∑

k>0

z−km

k

∂

∂tk

)(

∑

l>0

z−lm

l

∂

∂tl

)

log τ. (117)

Since am = 1/w(zm), the l.h.s. of this expression is noth-ing but an exponent of the growth probability in the

17

w-plane (80). Thus, taking into account the Richard-son’s evolution of the external harmonic moments (14),provided that a single source operates during each timeinterval, one can recast (80) in the form:

P (kclm+1) = Nm exp

− 1

Km~

∑

k,l>0

∂2 log τm∂tk∂tl

∆tk∆tl

,

(118)where by ∆tk = Km~ z−k

m /k we denoted a variation ofthe harmonic moments due to the active source. Thefirst fundamental form in the exponent of (118) deter-mines the metrics on the infinite dimensional Hermitianmanifold of external harmonic moments. The Hermitianmetric on this space is Kahler, and log τ is the Kahlerpotential [39]:

gkl =∂2 log τ

∂tk∂tl. (119)

The growth of the domain, when the tau-functionchanges with time, determines certain geometric flowon the manifold of harmonic moments. However, de-tailed analysis of these aspects of the stochastic Lapla-cian growth is beyond the scope of this paper.

IV. HAMILTONIAN STRUCTURE OF THE

BOUNDARY DYNAMICS

In this section we prove the Hamiltonian structure ofthe interface dynamics in the Laplacian growth. Let usbriefly recall the required elements of the classical me-chanics. Suppose the 2N -dimensional phase space isspanned by the canonical coordinates, qn and pn, wheren = 1, 2, . . . , N . The dynamical system is called Hamil-tonian, if its time evolution is governed by the Hamilton’sequations:

dpn

dt= − ∂H

∂qn,

dqndt

=∂H∂pn

. (120)

where the Hamiltonian, H = H(p, q, t), is a scalar func-tion of the canonical coordinates and time, correspondingto the total energy (the sum of the kinetic and potentialenergy) of the system. The Hamilton’s equations (120)can be obtained from the Hamilton’s principle. Con-sider the variation of the so-called phase-space action,S[q,p] =

∫ t2t1(∑

n pnqn − H)dt, w.r.t. qn and pn, suchthat the boundary conditions of the coordinate qn arefixed, δqn(t1) = δqn(t2) = 0, while nothing is requiredfor the momentum pn at the ends. Then, the variationyields the Hamilton’s equations as the extremal condi-tion.The Hamilton’s equations of motion are equivalent to

the Euler-Lagrange equation, which governs the timeevolution of the system in the Lagrangian approach.The configuration space in the Lagrangian formalism is

spanned by the set of generalized coordinates qn andvelocities qn. The two approaches are related by theLegendre transform, H(q,p, t) =

∑

n pnqn − L(q, q, t),where the canonical momenta are calculated by differ-entiating the Lagrangian with respect to the velocities,pn = ∂L/∂qn. The Euler-Lagrange equations of mo-tion follows upon setting the variation of the action func-tional, S[q, q] =

∫ t2t1

L(q, q, t) to zero, provided that theboundary conditions for the generalized coordinate arefixed:

d

dt

∂L∂qn

=∂L∂qn

. (121)

Now we are ready to consider the boundary’s dynamicsin the Laplacian growth problem. Let us first introducethe configuration space of the dynamical system. Thegeneralized coordinate q(ξ, t) will be identified with alength of the pathline, i.e. the trajectory that the bound-ary’s segment follows during the time evolution. Thecoordinates q(ξ, t) are labeled by the continuous indexξ ∈ Γ(t), and, therefore, the configuration space is theinfinite-dimensional function space. The pathlines de-pend on the full time-history of the boundary’s dynam-ics, and their starting points lie on the unit circle, suchthat q(ξ, 0) = 0 identically. It is naturally to define thegeneralized velocities, q(ξ, t), as the normal velocities ofthe boundary segments. Therefore, the generalized coor-dinates in the configuration space are:

q(ξ, t) =

∫ t

0

Vn(ξ, t′)dt′, q(ξ, t) = Vn(ξ, t). (122)

To determine the Hamiltonian for this system we ad-dress to electrostatic interpretation of growth probabil-ities (see discussion after (81)). The probability of thelayer takes a form of the Boltzmann weight, which isdetermined by the electrostatic energy of interaction be-tween induced charges along the unit circle in the w-plane. This energy can be treated as an increment ofthe total energy (Hamiltonian) of the system per growthstep:

dH(t) = −M(t)∑

m=1

qm(t) log(

1− |am(t)|2)

dt. (123)

In the z-plane the same functional takes a rather differentform, e.g. from (65) it follows:

dH(t) =

M(t)∑

m=1

qm(t)

∮

Γ(t)

µt(ξ, zm) log

∣

∣

∣

∣

∂nGt(ξ, zm)

w′t(ξ)

∣

∣

∣

∣

dt,

(124)where we took into account that ∂nGt(ξ,∞) = −|w′

t(ξ)|for ξ ∈ Γ. As we will prove, the representations of dH(t)at different planes, (123) and (124), are related by theHamilton’s equation.

18

To get rid of excessive summations over the M sourcesduring each growth step, it will be assumed that eachlayer is generated by a single source only. However, theresult can be generalized to the case of M sources.It will be shown, that canonical momentum can be re-

lated with the electrostatic potential of the domain, if toassume that D+ is filled homogeneously with an electriccharge of unit density. Let us introduce the potential ofthe layer:

Φl(z) = − 2

π

∫

l

d2z′ log |z − z′|. (125)

It admits the following series expansions around zero andinfinity respectively:

Φ+l (z) = −vl0 + 2Re

∑

k>0

tlkzk,

Φ−l (z) = −2tl0 log |z|+ 2Re

∑

k>0

vlkkz−k,

(126)

where vk and tk are the harmonic moments of thelayer (110). One can rewrite the harmonic extensionof the boundary function log |∂nG(ξ, zm)| to the pointzm ∈ D in terms of the potential of the layer. Takinginto account (86) and expanding the logarithm in 1/zm,we obtain:

∮

Γ

µ(ξ, zm) log |∂nG(ξ, zm)| = −G(zm,∞)−

− log |zm|+∮

Γ

µ(ξ, zm)Re∑

k>0

z−km

kξk. (127)

Using (111) and (112), we transform (127) to the form:

∮

Γ

µ(ξ, zm) log |∂nG(ξ, zm)| = π

2qmdt

∮

Γ

µ(ξ, zm)Φ−l (ξ).

(128)As for the denominator of the logarithm in (124) it can

be related with the Green’s function of the domain. Fromthe definition of the Green’s function (8), the followingidentity follows:

log

∣

∣

∣

∣

w(ξ)− w(zm)

ξ − zm

∣

∣

∣

∣

= G−(ξ, zm)+

+ log |w(zm)|+ log

∣

∣

∣

∣

∣

1− 1

w(ξ)w(zm)

∣

∣

∣

∣

∣

, (129)

where we used that log |w(ξ)| = 0 for ξ ∈ Γ. The har-monic extension of (129) from the boundary, ξ ∈ Γ, tothe point zm ∈ D reads:

∮

Γ

µ(ξ, zm) log |w′(ξ)| =∮

Γ

µ(ξ, zm)G−(ξ, zm)+

+ log |w(zm)|+ log(

1− |am|2)

, (130)

where am = 1/w(zm). Thus, using (128) and (130) we re-cast the increment of the Hamiltonian (124) in the form:

dH(t) = −qm(t) log(

1− |am(t)|2)

dt−

− qm(t)dt

∮

Γ(t)

µ(ξ, zm)(

G−(ξ, zm) + log |w(zm)|)

+

+π

2

∮

Γ(t)

µ(ξ, zm)Φl(t)(ξ). (131)

where the line integral along the boundary of the do-main is equivalent to the summation over continuous sub-script ξ ∈ Γ, which labels the boundary’s segments. Ther.h.s. of (131) resembles the differential of a certain func-tion of several variables. Indeed, the potential of thelayer l(t) = D(t) \ D(t − δt) can be treated as a dif-ference of the potentials of the corresponding domains,i.e. Φl(t) = ΦD(t) − ΦD(t−δt) ≡ dΦD(t). Besides, let ususe (63) to express the harmonic measure of the bound-ary through the normal boundary velocity, or generalizedvelocity: µ(ξ, zm) = q(ξ, t)|dξ|/qm. Since qdt = dq, wecan rewrite (131) as follows:

dH(t) = −qm(t) log(

1− |am(t)|2)

dt−

−∮

Γ(t)

|dξ|(

G−(ξ, zm) + log |w(zm)|)

dq(ξ, t)+

+

∮

Γ(t)

|dξ| q(ξ, t) d(

π

2qmΦ−

D(t)(ξ)

)

. (132)

The Hamiltonian should be a scalar function of thecoordinates q(ξ, t) and momenta p(ξ, t), such that itsdifferential can be written in the form dH(p, q, t) =∂tHdt + ∂pHdp + ∂qHdq. Thus, we can identify thecanonical momentum with the potential at the interface:

p(ξ, t) =π

2qmReΦ−

D(t)(ξ). (133)

Besides, we also obtain the equations:

∂H∂t

= −qm log(

1− |am|2)

,∂H∂p

= q(ξ, t),

∂H∂q

= −(

G−D(t)(ξ, zm) + log |wt(zm)|

)

.

(134)

Suppose the interface dynamics is Hamiltonian, i.e.the momentum obeys the Hamilton’s equation, namely,p = −∂H/∂q. Then, taking into account the the def-inition of the canonical momentum (133) and the lastequation in (134), we obtain the Laplacian growth equa-tion (42), which appears to be Hamilton’s equation forthe interface dynamics. Thus, we showed that the bound-ary’s dynamics is Hamiltonian and the time dependentHamiltonian can be written in the form:

H(t) = −∫ t

0

dt′M(t′)∑

m=1

qm(t′) log(

1− |am(t′)|2)

. (135)

19

Let us mention a rough analogy with quantum mechan-ics. For the time independent Hamiltonian, the quantumsystem can stay in a single state indefinitely long. Tran-sitions between different states, or time evolution of ob-servables, are possible if the additional time-dependentperturbation is applied to the quantum system. Thesame applies to the stochastic Laplacian growth, if oneidentifies the Hilbert space of the theory with the equiv-alence classes of grown patterns, such that the clusters,which differ by area only (while the other external har-monic moments are the same), are equivalent. Thus,the Hilbert space has a projective space structure. Thegrowth of the cluster, governed by a single hydrodynami-cal source at infinity, keeps the system in the same state.Therefore, there should be the time-independent Hamil-tonian H0, associated with this process. Transitions be-tween states (equivalence classes of patterns) occur dueto time-dependent perturbations, which change the har-monic moments of the cluster. The associated time-dependent Hamiltonian, which is commonly denoted byHint(t) in the quantum mechanical problems, is (135).

V. POSSIBLE CONNECTION TO THE

LIOUVILLE FIELD THEORY

As a final matter we address a possible relation be-tween the growth probabilities of layers in stochasticLaplacian growth and the semi-classical limit of corre-lation functions in the Liouville conformal field theory.We start from a simple observation, that is the Robin’sfunction [36],

G−(w,w) = − log |1− ww|, (136)

for the Dirichlet problem inside the unit disk in the w-plane, satisfies the classical Liouville equation:

∂∂G−(w, w) = e2G−(w,w) (137)

In differential geometry the Liouville equation appearsas a partial differential equation for the conformal fac-tor, expϕ(w, w), of a metric ds2 = expϕ(w, w)|dw|2 ona two-dimensional surface of constant Gaussian curva-ture. In particular, if ϕ(w, w) = 2G−(w, w), the Liou-ville equation (137) describes the geometry of the so-called Lobachevskiy plane, or pseudosphere, which is atwo-dimensional surface with a constant negative curva-ture Rds2 = −4. It can be realized as the Poincare diskmodel, such that the points of the geometry are insidethe unit disk B = w ∈ C : |w| < 1. The straight linesconsist of circle arcs inside the disk, that are orthogonalto the boundary of the disk. The points of the unit circle,|w| = 1, are called absolute since they are infinitely dis-tant from all other points in B. The associated metricsds2 = expϕ(w, w)|dw|2 is given by the solution to theLiouville equation (137):

ϕ(w, w) = − log(1 − ww)2. (138)

Since the Liouville field, ϕ(w, w) appears as the confor-mal factor of a metric tensor under holomorphic coordi-nate transformations, w → z = f(w), it transforms as

ϕ→ ϕ(w, w) = ϕ(z, z) + 2 log |f ′(w)|. (139)

By virtue of this transformation law one can easily relatethe probabilities of the layers in different planes, e.g. (76)and (80), directly. In terms of the Liouville field, theprobability of the layer in the w-plane (80) appears as aproduct of conformal factors:

P (kcli ) = Ni

M∏

m=1

e−(Km/2)ϕ(am,am). (140)

This expression can be treated as the semi-classicallimit of a certain correlation function in the Liouville fieldtheory on the pseudosphere. To be more specific, we needto recall briefly the required elements of this theory (fordetails see [37]). The Liouville equation (137) appears asthe Lagrange-Euler equation for the Liouville action:

SL[ϕ] =1

8πb2

∫

|w|<1

(

2|∂ϕ|2 + 8eϕ)

d2w + S, (141)

where the integral is taken over the unit disk and by S wedenoted the contribution of the boundary to the action.An explicit form of S is not important at this point, butcan be fixed by requirement of finiteness of the actionand invariance under conformal transformations (139).The coefficient b in (141) is an important parameter ofthe Liouville field theory. Note, that we consider the so-called classical limit, b→ 0, of the Liouville theory, suchthat the action (141) and the field, ϕ(w), are assumed tobe renormalized in an appropriate way.The Liouville action determines the correlation func-

tions of the exponential fields,

⟨

M∏

m=1

eαm

bϕ(amam)

⟩

≡∫

[dϕ]e−SL[ϕ]M∏

m=1

eαm

bϕ(am,am),

(142)where the parameters αm label different conformalclasses of exponential operators [38]. Let us considerthe classical limit of the correlation functions. For smallb one can use the saddle-point approximation to evalu-ate the path-integral in (142). However, the scaling ofαm’s with b should be specified first. Since the Liou-ville action scales like b−2, from (142) it follows that twodifferent classes of primary operators exist. The oper-ators with αm ∼ b−1 are called the “heavy” Liouvilleprimary fields, because their conformal dimensions di-verge as b−2. These operators non-trivially effect on thesaddle point. One can also consider the so-called “light”operators with α = µb (where µ is a constant) and fixedconformal dimension for b → 0. In the classical limitthe “light” fields influence neither the classical solution

20

nor the one-loop correction. In the case of the pseudo-spherical geometry (138), semiclassical behaviour of thecorrelation function of “light” fields reads:

⟨

M∏

m=1

eµmϕ(am,am)

⟩

≈ e−SL[ϕcl]M∏

m=1

eµmϕcl(am,am).

(143)The exact value of SL[ϕcl] is not important at thispoint. Leaving aside the normalization problem, we no-tice a similarity between the statistical weights of thelayers (140) and the correlation functions of the “light”fields (143), provided the identification µm = −Km/2.Thus, the growth probability of the layer in the stochas-tic Laplacian growth is given by a semi-classical limit ofthe multi-point correlation functions of “light” exponen-tial fields in the Liouville field theory on a pseudosphere.To summarize the results of this section we note, that

similarity between growth probabilities and certain cor-relation functions in the Liouville theory is not a co-incidence. The quantum Lioville field theory appearsquite naturally in the context of the stochastic Lapla-cian growth. We will address this problem in detail insubsequent publications.

CONCLUSIONS

In this paper we have considered the determinis-tic (classical) limit of the discrete stochastic Laplaciangrowth model, which reveals a strong connection be-tween Laplacian growth and diffusion limited aggrega-tion. We related stocastic growth with generation ofvirtual sources, which govern the interface dynamicsand correspond to the quadrature points of the domain.Clearly, if the virtual sources appear sufficiently close tothe boundary instead of smooth layers, one can observegeneration of “bumps”, i.e. localized bell-shaped distri-butions of attached particles at the boundary. This sim-ple observation unites the stochastic Laplacian growthmodel with the Hastings-Levitov approach to pattern for-mation using conformal maps [11]. Simple probabilisticarguments allow us to introduce the probability of thegenerated patterns, so providing an analytic frameworkto study stochastic pattern formation.The growth probability of a single layer has a form of

Kullback-Leibler entropy, and can be rewritten as certainfunctional on the layers, which have remarkable relationswith the tau-function for the boundary curve. This rela-tion allows us to relate stochastic Laplacian growth withthe theory of normal random matrices. For the classi-cal LG this relation has been noticed in [21] and furtherinvestigated in a number of papers [31, 34, 35]. Unfortu-nately, the multi-point density correlation functions com-puted in the random matrix framework [40] do not repro-duce the correct Hausdorff dimension of the grown clus-ters. Keeping aside possible speculations on this distinc-

tion, we note that the normal random matrices appearin the stochastic Laplacian growth model in a slightlydifferent context. Due to peculiarities of the stochasticgrowth process, such that domain grows by adding layers,the probabilities associated with the layer determine thedistribution function of eigenvalues of the normal ran-dom matrices inside the layer. As a consequence, theeigenvalues belonging to different layers do not interact.The distribution function of eigenvalues associated withthe final domain appears as a product of the distributionfunctions associated with the layers. Therefore computa-tion of the multi-point density correlation functions forthe matrix model with the eigenvalues distributed insidethe tubular-like neighborhood of some curve seems to bean important step for determination of the Hausdorff di-mension of the interface.

Remarkably, we managed to relate the layers proba-bilities (118) with the geometric characteristics (metrictensor (119)) of the infinite-dimensional Kahler manifold,spanned by the external harmonic moments of the grow-ing domain. The evolution of the domain generates cer-tain geometric flow on the manifold, as the tau-functionchanges with time.

We proved the Hamiltonian structure of the interfacedynamics, so that the famous Laplacian growth equa-tion (more precisely (42)) turns out to be the Hamilton’sequation for this dynamical system. We also determinedthe time-dependent Hamitonian, which generates transi-tions between different states (equivalence classes of con-tours) in the Hilbert space of the theory. To completethe description of the Laplacian growth as a Hamilto-nian system, the time-independent Hamiltonian shouldbe determined.

Finally, the observed relation between growth proba-bilities of layer and certain correlation functions in theLiouville field theory in a pseudosphere awaits furtherclarification. A very few remarks on the related subjectscan be found in literature. For example, in [34] the nextorder corrections (namely F1) to the free energy in thelarge N expansion for the two-dimensional Dyson gas(i.e. the complex β-ensemble) had been related with thespectral determinant of the Laplace-Beltrami operatorfor the unit disk. This observation suggests interestinglinks with the Liouville field theory. We believe, that ourapproach to the stochastic Lapacian growth can elucidatethis topic in many details.

[1] T. Witten and L. Sander, Diffusion-limited aggregation,Phys. Rev. B 27, 5686 (1983);

[2] T.C. Halsey, Physics Today 53, 36 (2000);[3] D. Bensimon, L. Kadanoff, S. Liang, B. Shraiman and

C. Tang, Viscous flows in two dimensions, Rev. Mod.Phys. 58, 977 (1986);

[4] B. Shraiman and D. Bensimon, Singularities in nonlocal

21

interface dynamics, Phys. Rev. A 30, 2840 (1984);[5] S. Howison, Fingering in Hele-Shaw cells, J. Fluid Mech.

167, 439 (1986);[6] M. Mineev-Weinstein and S. Dawson, Class of nonsingu-

lar exact solutions for Laplacian pattern formation. Phys.Rev. E 50, R24(R) (1994);

[7] P. Saffman and G. Taylor, The penetration of a fluid into

a porous medium of Hele-Shaw cell containing a more vis-

cous fluid Proc. R. Soc. London Ser. A 245, 312 (1958);[8] B.I. Shraiman, Velocity Selection and the Saffman-Taylor

Problem, Phys. Rev. Lett. 56, 2028 (1986); D.C. Hongand J.S. Langer, Analytic Theory of the Selection Mech-

anism in the Saffman-Taylor Problem Phys. Rev. Lett.56, 2032 (1986); R. Combescot et al., Shape Selection

of Saffman-Taylor Fingers Phys. Rev. Lett. 56, 2036(1986); S. Tanveer, Analytic theory for the selection of

a symmetric Saffman–Taylor finger in a Hele–Shaw cell,Phys. Fluids 30, 1589 (1987);

[9] M. D. Kruskal and H. Segur, Asymptotics beyond All Or-

ders in a Model of Crystal Growth, Stud. Appl. Math. 85,129 (1991);

[10] M. Mineev-Weinstein, Selection of the Saffman-Taylor

Finger Width in the Absence of Surface Tension:

An Exact Result, Phys. Rev. Lett. 80, 2113 (1998);arXiv:patt-sol/9705004;

[11] M. Hastings and L. Levitov. Laplacian growth as one-

dimensional turbulence, Physica D 116, 244 (1998);arXiv:cond-mat/9607021;

[12] A. Hurwitz and R. Courant, Vorlesungen uber all-

gemeine Funktionentheorie und elliptische Funktionen.

Herausgegeben und erganzt durch einen Abschnitt uber

geometrische Funktionentheorie, Springer-Verlag, 1964(Russian translation, adapted by M.A. Evgrafov: The-ory of functions, Nauka, Moscow, 1968);

[13] S. Richardson, Hele-Shaw flows with a free boundary pro-

duced by the injection of fluid into a narrow channel, J.of Fluid Mech. 56, 609 (1972);

[14] P. S. Novikov Inverse Potential Problem, Doklady Acad.Sci USSR. 56, 609 (1938);

[15] P. Davis, The Schwarz function and its applications, Buf-falo, N.Y.: The Math. Association of America, 1974;

[16] G. Herglotz, Uber die analytische Fortsetzung des Poten-

tials ins Innere der anziehenden Massen, Preisschr. derJablonowski gesselschaft 44, 1914.

[17] D. Aharonov and H. Shapiro, Domains on which analytic

functions satisfy quadrature identities, J. Anal. Math. 30,39 (1976);

[18] B. Gustafsson, Quadrature identities and the schottky

double, Acta Applicandae Mathematicae 1, 209 (1983);[19] P. Etingof and A. Varchenko, Why does the boundary of

a round drop becomes a curve of order four, UniversityLecture Series, 3, American Mathematical Society, Prov-idence, RI, 1992;

[20] M. Mineev-Weinstein, P. Wiegmann and A. Zabrodin. In-tegrable structure of interface dynamics, Phys. Rev. Lett.84, 5106 (2000); arXiv:nlin/0001007 [nlin.SI];

[21] I. Kostov, I. Krichever, M. Mineev-Weinstein, P. Wieg-mann and A. Zabrodin, τ -function for analytic curves,Random matrices and their applications, MSRI pub-lications, 40, Cambridge University Press, 285 (2001);arXiv:hep-th/0005259;

[22] J. Garnett and D. Marshall, Harmonic measure, Cam-bridge University Press, 2005;

[23] P. Pelce, Dynamics of Curved Fronts, Academic Press,

San Diego, 1988;[24] S. Kullback and R.A. Leibler, On Information and Suffi-

ciency, Ann. of Math. Statistics 22, 79, MR 39968 (1951);[25] I. Gruzberg, D. Leshchiner and M. Mineev-Weinstein,

Unpublished;[26] Yu. Vinogradov and P. Kufarev, On some particular so-

lutions of the problem of filtration (N.S.), Dokl. Akad.Nauk SSSR 57, 335 (1947);

[27] L. Galin, Unsteady filtration with a free surface, Dokl.Akad. Nauk. S.S.S.R., 47, 246 (1945);

[28] P. Polubarinova–Kochina, On a problem of the motion

of the contour of a petroleum shell Dokl. Akad. Nauk.S.S.S.R., 47, 254 (1945);

[29] Ar. Abanov, M. Mineev-Weinstein and A. Zabrodin,Multi-cut solutions of Laplacian growth, Physica D 238,1787 (2007); arXiv:0812.2622 [nlin.SI];

[30] D. Khavinson and H. Shapiro, The Schwarz Potential in

Rn and the Cauchy Problem for the Laplace Equation,Royal Inst. of Tech, Stockholm, 1989;

[31] O. Agam, E. Bettelheim, P. Wiegmann and A. Zabrodin,Viscous fingering and a shape of an electronic droplet in

the Quantum Hall regime; Phys. Rev. Lett. 88, 236801(2002); arXiv:cond-mat/0111333;

[32] L.-L. Chau and Y. Yu, Unitary polynomials in normal

matrix models and wave functions for the fractional quan-

tum Hall effects, Phys. Lett. 167A, 452 (1992);[33] L.-L. Chau and O. Zaboronsky, On the structure of

Normal Matrix Model, Commun. Math. Phys. 196, 203(1998), arXiv:hep-th/9711091;

[34] A. Zabrodin and P. Wiegmann. Large N expansion

for the 2D Dyson gas, J.Phys. A39, 8933 (2006);arXiv:hep-th/0601009;

[35] R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodinand P. Wiegmann, Normal random matrix ensemble

as a growth problem Nucl. Phys. B 704, (2005) 407;arXiv:hep-th/0401165;

[36] M. Flusher and B. Gustafsson, Vortex motion in two-

dimensional hydrodynamics, Royal Inst. Techn. Stock-holm TRITA-MAT-9J-MA-02 (1997);

[37] A. Zamolodchikov and Al. Zamolodchikov, Liouville field

theory on a pseudosphere, arXiv:hep-th/0101152;[38] A. Zamolodchikov and Al. Zamolodchikov, Struc-

ture Constants and Conformal Bootstrap in Liou-

ville Field Theory, Nucl. Phys. B 477, 577 (1996);arXiv:hep-th/9506136;

[39] L. Takhtajan, Free bosons and tau-functions for compact

Riemann surfaces and closed smooth Jordan curves I.

Current correlation functions, Lett. Math. Phys. 56, 181(2001); arXiv:math/0102164 [math.QA];

[40] P. Wiegmann and A. Zabrodin, Large scale correlations

in normal and general non-Hermitian matrix ensembles,J. Phys. A 36, 3411 (2003); arXiv:hep-th/0210159

[41] the potential equals zero on it[42] Here the label n goes from 1 to N , while i – from 1 to

T/δt, where T is a total time of the growth process.[43] It is supposed that the distant source is located at infin-

ity.[44] It is convenient to use discrete notation for time steps as

long as we consider the layer’s probabilities.[45] Without this factor the probability of the classical tra-

jectory equals 1 identically.[46] the next order terms, O(log km(ξ, i)), of the Stirling ap-

proximation are omitted in the exponent.[47] In fact, it becomes a cut-off dependent constant