Limit theorems for Levy walks in d dimensions: rare and bulk fluctuations ∗

Itzhak Fouxon1,2,† Sergey Denisov3,4,‡ Vasily Zaburdaev3,5,§ and Eli Barkai1¶1 Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel

2 Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea3 Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod,

Gagarina Av. 23, Nizhny Novgorod, 603950, Russia4 Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine and

5 Max Planck Institute for the Physics of Complex Systems,Nothnitzer Strasse 38, D-01187 Dresden, Germany

We consider super-diffusive Levy walks in d > 2 dimensions when the duration of a single step,i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution ofinfinite variance and finite mean. We demonstrate that the probability density function (PDF)of the coordinate of the random walker has two different scaling limits at large times. One limitdescribes the bulk of the PDF. It is the d−dimensional generalization of the one-dimensional Levydistribution and is the counterpart of central limit theorem (CLT) for random walks with finitedispersion. In contrast with the one-dimensional Levy distribution and the CLT this distributiondoes not have universal shape. The PDF reflects anisotropy of the single-step statistics howeverlarge the time is. The other scaling limit, the so-called ’infinite density’, describes the tail of thePDF which determines second (dispersion) and higher moments of the PDF. This limit repeatsthe angular structure of PDF of velocity in one step. Typical realization of the walk consists ofanomalous diffusive motion (described by anisotropic d−dimensional Levy distribution) intermittedby long ballistic flights (described by infinite density). The long flights are rare but due to themthe coordinate increases so much that their contribution determines the dispersion. We illustratethe concept by considering two types of Levy walks, with isotropic and anisotropic distributions ofvelocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distribution thed−dimensional process can be reduced to one-dimensional Levy walk.

I. INTRODUCTION

It is a universal consequence of microscopic chaos thatthe velocity v(t) of a moving particle has a finite corre-

lation time [1, 2]. The particle’s displacement∫ t0v(t′)dt′

on large time scales can be considered as outcome of asequence of independent random ’steps’. A single stephere is a motion during the shortest time over which thevelocity of the particle is correlated. Velocity correlationsat different steps can be neglected and the integral overthe time interval [0, t] can be replaced with the sum of in-tegrals over disjoint steps. If variations of step durationsand velocity fluctuations can be neglected as well, we ar-rive at the standard random walk consisting of steps thattake the same fixed time. The distance covered duringsingle step is fixed but the direction of the step is random.It can be, for example, a step along one of the basis vec-tors when random walk is performed on a d−dimensionallattice or it can be a step in a completely random direc-tion in d−dimensional space as in the case of isotropicwalk. In many situations though the variation of thestep’s duration and velocity cannot be disregarded. A

∗Submitted to Smoluchowski’s special issue Gudowska-Novak, Lin-denberg, Metzler, Editors.†Electronic address: itzhak8@gmail.com‡Electronic address: sergey.denisov@physik.uni-augsburg.de§Electronic address: vzaburd@pks.mpg.de¶Electronic address: eli.barkai@biu.ac.il

famous example is Levy walks (LWs) [3–27] that belongto a more general class of stochastic processes called con-tinuous time random walks (CTRWs) [3, 4]. In CTRWsit is not only the direction of the displacement during onestep that is random (as in the standard random walk) butalso the length of the step and the time that it takes. Thisflexibility allows to cover a large number of real-life sit-uations including dynamics of an ordinary gas moleculewhen both the time between consecutive collisions andvelocity of the molecule vary.

In ideal gas the probability of large (much larger thanthe mean free time) time between the collisions is negli-gibly small so the probability density function (PDF) ofthe step duration decays fast for large arguments. Thisis not the case for the so-called Lorentz billiard [2], inwhich the particle moves freely between collisions withscatterers arranged in a spatially periodic array. In thecase without horizon, when infinitely long corridors be-tween the scatterers are present, the particle can fly freelyfor a very long time if its velocity vector aligns close tothe direction of the corridor. The distribution of timesbetween consecutive collisions has a power-law tail andinfinite variance [31]. The dynamics of a particle canbe reproduced with a Levy walk process to great detail[32, 33]. Levy walks were found in diverse real-life pro-cesses including the spreading of cold atoms in opticallattices, animal foraging, and diffusion of light in disor-dered glasses and hot atomic vapors (for more examplessee a recent review [20] and references therein). Despiteof these advances and new experimental findings, the the-ory of LWs remains mainly confined to the case of one-

2

dimensional geometry.In this work we study d−dimensional Levy walks when

the duration τ of single steps is characterized by a PDFψ(τ) with power-law asymptotic form, ψ(τ) ∝ τ−1−α.The most interesting and practically relevant is the so-called ’sub-ballistic super-diffusive regime’ [20], 1 < α <2, when the mean step time 〈τ〉 is finite but the dispersion〈τ2〉 is infinite [28]. We study the PDF P (x, t) of thewalker’s coordinate x(t) at time t in this regime. Weshow that there are two ways of rescaling P (x, t) withpowers of time that produce finite infinite time limits.The two limiting distributions describe the bulk and thetails of P (x, t). Both distributions are sensitive to themicroscopic statistics of the velocity of walkers and aremodel specific.

We consider a random walk in Rd with a PDF F (v)of the single step velocity that obeys F (−v) = F (v), cf.[29, 30]. Thus the process is unbiased and the averagedisplacement is zero. In the case of the sum of a largenumber of independent and identically-distributed (i. i.d.) random variables with finite dispersion there is awell-known scaling,

limt→∞

td/2P(x√t, t)

= g(x), (1)

where g(x) is a Gaussian distribution [47]. A stronger,large deviation limit tells that

limt→∞

1

tlnP (tx, t) = s(x), (2)

where the convex function s(x) is known as large devia-tions, or entropy, or rate, or Kramer’s function [34, 35];see [36] for simple derivation. This limit describes ex-ponential decay of the probability of large deviations offinite-time value x(t)/t from its infinite time limit fixedby the law of large numbers, limt→∞(1/t)x(t) = 0. Itcorresponds to the Boltzmann formula and tells thatthe PDF of macroscopic thermodynamic variable (rep-resentable as the sum of large number of independentrandom variables) is exponential of the entropy whosemaximum’s location gives the average. The coefficientsof the quadratic expansion near the maximum character-ize thermodynamic fluctuations [35]. Thus in the case offinite dispersion there is one universal scaling with thelimiting distribution (the special case of finite dispersionbut power-law tail, α > 2, produces different limitingdistribution [37]; we do not consider this case here).

There are two different scalings that were found in thecase of one-dimensional super-diffusive LWs [17, 18]. Oneof them is a continuation of the central limit theorem(CLT) to the case of i. i. d. random variables with in-finite dispersion, the so-called generalized central limittheorem (gCLT) [49]. The corresponding distribution isknown as the celebrated Levy distribution. We first gen-eralize this scaling to the d−dimensional case and findsignificant difference from the one-dimensional case andordinary d−dimensional random walks. For both theone-dimensional LW and d−dimensional random walks

there is universality: the form of the scale-invariant PDFdoes not depend on details of the single step statistics(for instance, ordinary two-dimensional random walks ontriangular and square lattices are described by the sameisotropic Gaussian PDF in the limit of large times [39]).In the case of d−dimensional LWs the anisotropy of thesingle step statistics is imprinted into the statistics of thedisplacement – no matter how large the observation timet is; see the discussion of the particular case of d = 2 inRef. [22].

In the language of field theory, ordinary random walksare renormalizable: in the long-time limit the informa-tion on macroscopic structure of the walks is reduced tofinite number of constants [39]. Thus the Gaussian PDFof the sum of large number of i. i. d. random variables isfully determined by the mean and the dispersion of thosevariables and their total number. The rest of the infor-mation on the statistics of these variables is irrelevant –when the Gaussian bulk of the PDF is addressed. In con-trast, the entropy function of the large deviations theoryis the Legendre transform of the logarithm of the charac-teristic function of the random variable in the sum. Thuslarge deviations description is sensitive to the details ofstatistics of one step of the walk. We conclude that theGaussian bulk of the PDF of the sum of large number ofi. i. d. random variables with finite dispersion is deter-mined by a finite number of constants characterizing thestatistics of the single step but the tail of the PDF is not.

In this work we show that the PDF of a d - dimen-sional LW is not universal and cannot be specified witha finite number of constants – already in the bulk – ifthe statistics of the velocity of single steps, given by thePDF F (v), is anisotropic. To stress this fact we callthe corresponding distributions ’anisotropic Levy distri-butions’. In contrast, if the velocity statistics is isotropic,for example, F (v) is given by the uniform distributionon the surface of the d-dimensional sphere of the radiusυ0 = const (see Fig. 1a), the description of the processcan be reduced to one-dimensional case and the bulk ofthe corresponding PDF is fully determined by a finitenumber of constants which can be calculated from F (v).Thus in the anisotropic case there is a dramatic differencebetween the bulk of the PDFs of a LW and the randomwalk with finite dispersion of the single step duration.

Furthermore, the tail of the PDF P (x, t) have cer-tain distinct features in the regime of sub-ballistic super-diffusion. The rare events when the walker performs ex-tremely long ballistic flights have a substantial impacton the total displacement of the walker even in the limitof long times. The probability of long ballistic steps isnot negligibly small and single ballistic steps could beseen in a single trajectory of the walker for any time t.Such steps form the outer regions of the PDF and in one-dimensional case they are described with the so-called ’in-finite density’, where the word ’infinite’ refers to the non-normalizable character of this function [17, 18, 50, 51].

The two limit distributions describe the bulk and thetail of the LW’s PDF, respectively. The bulk is formed

3

FIG. 1: Examples of Levy walks in three dimensions. A walker has a constant (by absolute value) velocity υ0 = |v| = 1.When performing uniform Levy walk (a), the walker, after completing a ballistic flight, instantaneously selects a random timeτ for a new flight and randomly chooses a flight direction (it can be specified by a point on the surface of the unit sphere).The velocity PDF F (v) is described by the uniform distribution over the unit sphere’s surface. In the case of anisotropic XY ZLevy walk (b), the walker is allowed to move only along one axis at a time. After the completion of a flight, the walker selectsa random time τ for a new flight, one out of six directions (with equal probability) and then moves along chosen direction. Thevelocity PDF F (v) in this case is six delta-like distributions located at the points where the unit sphere is penetrated by theframe axes. The parameter α is 3/2.

by the accumulation of typical (most probable) steps.They are responsible for a diffusive motion (albeit al-ready anomalous one). In contrast, the PDF’s tails areformed by long ballistic flights and they are described bythe infinite density. These flights are rare steps where thewalker moves for a long (i. e., comparable to the totalobservation time t) time without changing its velocity.In the case of the Lorentz billiard this is the situationwhen the velocity vector of the particle aligns close tothe direction of one of the ballistic corridors [32]. Thoughthis happens relatively rarely, the distance covered by thewalker during a such flight is so large that these flightsgive finite contribution to the probability that the walkerdisplacement after the time t is of the order v0t. Whenthe probability of long flights is, for example, exponen-tially small (as in the case of the standard random walks)the contribution of the flights can be neglected. This isnot, however, the case of the LWs with power-law asymp-totic of ψ(τ), as we demonstrate in this paper.

The paper is organized as follows. In Section II we in-troduce the basic definitions and the tool of the study -the Fourier-Laplace transform of the PDF of the walker’scoordinate. In the next Section we provide complete so-lution for the case of isotropic statistics of velocity of thewalker. Central result of our work - the anisotropic CLTfor the bulk of the PDF is derived in Section IV. Thenext Section describes universality of the tail of this non-

universal bulk that helps finding low moments of the dis-tance from the origin. Section VI provides the other lim-iting theorem on infinite density that provides the tail ofthe distribution. The next Section provides detailed formof the moments of arbitrary order including anomaly ingrowth due to anisotropy. In Section VIII we provide thetail of the PDF and Conclusions resume our work.

II. FOURIER-LAPLACE TRANSFORM OF THEPDF P (x, t)

In this Section we specify the considered random pro-cess and introduce the main tool of the analysis on whichall further results rest. This is the Fourier (in space) -Laplace transform (in time) of P (x, t).

We consider a LW as an infinite sequence of flights(steps) of random duration τi where i is the flight’s in-dex in the chronologically ordered sequence. The processstarts at time t = 0 at the point x = 0. The velocity ofthe walker during a flight is a random vector vi which re-mains constant during the flight. Upon the completion ofthe flight both the velocity vi+1 and the duration τi+1 ofthe next flight are randomly chosen, by using PDFs F (v)and ψ(τ), respectively. The number of flights performedduring the observation time t, N(t), is a random number

constrained by t =∑N(t)i=1 τi + τb , where τb = t − tN is

4

so-called the backward recurrence time [38].The simplest model is d−dimensional ’Levy plotter’,

the product of d independent one-dimensional walksalong the basis vectors which span Rd. The PDF ofthis process is the product of the corresponding one-dimensional PDFs [17, 18, 20]. This case demands nofurther calculations so we next consider non-trivial set-ups.

We consider two intuitive models, the uniform LW andanisotropic XY Z... LW. In the uniform model F (v) isspecified by the uniform distribution on the surface ofd−dimensional unit sphere so velocity has fixed mag-nitude 1; see Fig. 1a. As we demonstrate in the nextsection, in many respects this model can be reduced tothe one-dimensional case. In the anisotropic XY Z...model particle moves along one of the d basis vectorsat a time; see Fig. 1b. The analysis we present be-low is valid for any PDF F (v) obeying the symmetryF (−v) = F (v). As an illustration, we consider a par-ticular type of LWs in Rd with factorized velocity distri-bution F (v) = Fv(|v|) · Fd(v/|v|). In this product PDFfirst multiplier controls the absolute value of the velocity[the simplest choices is Fv(|v|) = δ(|v|−v0)] while secondmultiplier is governs the direction statistics of steps. APDF Fd(v/|v|) is a subject of directional statistics [30]and can be specified with a probability distribution onthe surface of the (d− 1) dimensional unit sphere in Rd.For example, in R3 the continuous transition from theisotropic model to the XY Z LW can be realized withsix von Mises-Fisher distributions [30] (centered at thepoints where the axes pierce the unit sphere) by tun-ing the concentration parameter of the distributions fromzero to infinity. We demonstrate in the following sectionsthat different statistics enter the PDF P (x, t) throughthe moments 〈(k · v)2n〉. In particular, for the XY Z..model we have

〈(k · v)2n〉 =

∑di=1 k

2ni

d(v0)

2n. (3)

In [22] we discuss physical models belonging to differentclasses of symmetry, e. g. the Lorentz gas with infinitehorizon belongs to the XY Z.. class.

The remaining PDF that defines the walk process isψ(τ). Below we consider ψ(τ) that has the tail

ψ(τ) ∼ A

Γ(−α)τ−1−α, 1 < α < 2, (4)

where A > 0 and Γ(x) is the gamma function (observethat Γ(−α) > 0 when 1 < α < 2). The factor Γ(−α) isintroduced in order to make the Laplace transform ψ(u)of ψ(τ)

ψ(u) =

∫ ∞0

exp[−uτ ]ψ(τ)dτ, (5)

to have small u behavior [that is determined by the tailof ψ(τ)],

ψ(u) = 1− 〈τ〉u+Auα + . . . , (6)

where 〈τ〉 =∫∞0tψ(t)dt is the average waiting time and

dots stand for higher-order terms.We use k and u to denote coordinates in Fourier and

Laplace space, respectively. By explicitly providing theargument of a function, we will distinguish between thenormal or transformed space, for example ψ(τ) → ψ(u)and g(x)→ g(k).

The lower limit of τ for which Eq. (4) holds depends onthe considered model. For instance the inverse gammaPDF,

ψ(τ) =2τ−5/2√

πexp

[−1

τ

], (7)

the tail described by Eq. (4) holds at τ � 1 with α = 3/2and A = 8/3. The corresponding Laplace pair obeys,

ψ(u) =[1 + 2

√u]

exp[−2√u]∼ 1− 2u+

8u3/2

3, (8)

that reproduces Eq. (6) where we use

〈τ〉 =

∫ ∞0

2τ−3/2dτ√π

exp

[−1

τ

]=

2Γ(1/2)√π

= 2. (9)

We introduce the key instrument of our analysis, theMontroll-Weiss equation. It provides with the Laplacetransform

P (k, u) =

∫ ∞0

exp[−ut]P (k, t)dt, (10)

of the characteristic function of the position x(t) of therandom walker at time t,

P (k, t) = 〈exp[ik · x(t)]〉 =

∫exp[ik · x]P (x, t)dx,(11)

in terms of averages over statistics of v and τ . We have

P (k, u) =

⟨1− ψ(u− ik · v)

u− ik · v

⟩1

1− 〈ψ(u− ik · v)〉,(12)

see Appendix A. Here the angular brackets denote theaveraging over the PDF F (v). The technical problem isto invert this formula in the limit of large time.

III. ISOTROPIC MODEL

In this Section we consider isotropic statistics of ve-locity where F (v) depends on |v| only. The magnitudeof velocity is a random variable drawn from the PDFSd−1v

d−1F (v) where Sd−1 = 2πd/2/Γ(d/2) is the areaof unit sphere in d dimensions (2π in d = 2 and 4π ind = 3). We show that the PDF P (x, t) of a LW in Rdcan be derived from one-dimensional distribution. Thusthe well-developed theory of one-dimensional LWs can beused for describing d-dimensional isotropic LWs.

We start with observation that for isotropic statistics ofv, the average of an arbitrary function h of k ·v depends

5

only on |k|; thus 〈h(k · v)〉 can be obtained by takingk = kx (where x is unit vector in x−direction),

〈h(k · v)〉 = 〈h(kvx)〉 =

∫ ∞−∞

h(kvx)F (vx)dvx,(13)

where F (vx) is the PDF of x−component of the velocity.It can be written in terms of the PDF F (v) = F (v) =Fd(v/|v|) which obeys the normalization,∫

F (v)dv = Sd−1

∫ ∞0

vd−1F (v)dv = 1. (14)

For d > 2 we have,

F (vx) =

∫δ(vx − v′x)F (v′)dv′∫

F (v′)dv′

=

∫∞|vx|(v

′)d−1F (v′)dv′∫ π0δ(vx − v′ cos θ) sind−2 θdθ∫∞

0(v′)d−1F (v′)dv′

∫ π0

sind−2 θdθ

=

∫∞|vx|(v

′)d−1F (v′)dv′∫ 1

−1 δ(vx − v′x)(1− x2)(d−3)/2dx

S−1d−1∫ 1

−1(1− x2)(d−3)/2dx

=2π(d−1)/2

Γ[(d− 1)/2]

∫ ∞|vx|

vd−2(

1− v2xv2

)(d−3)/2

F (v)dv. (15)

where we used∫ 1

−1(1 − x2)(d−3)/2dx =√πΓ[(d −

1)/2]/Γ(d/2) and Eq. (14). In the case of two dimen-sions θ varies between 0 and 2π not π but the calculationstill holds. Thus Eq. (15) provides the distribution ofx−component of velocity in arbitrary space dimensiond > 1.

Equation (15) can be simplified further in the caseof uniform model with velocity v0 where F (v) =

v1−d0 S−1d−1δ(v − v0). Integration in Eq. (15) gives

F (vx) = PSd/2−2,v0(vx),

where PSd/2−2,v0(v) is the (normalized) power semicirclePDF with range v0 and shape parameter d/2 − 2 thatvanishes when |v| > v0 and for |v| < v0 is given by [41]

PSd/2−2,v0(v) =Γ(d/2)√

πv0Γ[(d− 1)/2]

(1− v2

v20

)(d−3)/2

.(16)

The moments of this distribution read

〈|vx|γ〉 =vγ0Γ[(γ + 1)/2]Γ(d/2)√

πΓ[(d+ γ)/2]. (17)

Note that the ratio of gamma functions can be rewrit-ten as a product if d is an odd number. Finally, fromthis PDF we can derive the PDF and the moments forarbitrary F (v). For the PDF we find from Eqs. (15) and(16),

F (vx) =2πd/2

Γ(d/2)

∫ ∞|vx|

vd−1PSd/2−2,vx(v)F (v)dv, (18)

that can also be seen directly from the definition. Forthe moments, interchanging the order of integrations, weobtain from Eq. (15) the identity

〈|vx|γ〉 =

∫ ∞0

vd−10 Sd−1F (v0)dv0

[2π(d−1)/2

Γ[(d− 1)/2]∫|vx|γdvx

∫ ∞|vx|

vd−2(

1− v2xv2

)(d−3)/2δ(v − v0)

vd−10 Sd−1dv

].(19)

By noting that the term in brackets is the correspondingmoment of PSd/2−2,v0 , we find

〈|vx|γ〉=2π(d−1)/2Γ[(γ + 1)/2]

Γ[(d+ γ)/2]

∫ ∞0

vd+γ−1F (v)dv. (20)

For the Gaussian distribution F (v) =(2πv20)−d/2 exp

[−v2/2v20

]it gives

〈|vx|γ〉 =2γ/2Γ[(γ + 1)/2]vγ0√

π, (21)

that reproduces 〈v2x〉 = v20 when γ = 2. We observe that〈|v|γ〉 = Sd−1

∫∞0vd+γ−1F (v)dv so that Eq. (20) implies

the identity

〈|vx|γ〉 =Γ[(γ + 1)/2]Γ(d/2)

Γ[(d+ γ)/2]√π〈|v|γ〉. (22)

This formula is a consequence of the isotropy of the pro-cess and for any random vector x whose PDF dependson |x| only we have,

〈|xs|γ〉 =Γ[(γ + 1)/2]Γ(d/2)

Γ[(d+ γ)/2]√π〈|x|γ〉, (23)

where xs, s ∈ {1, 2, ..., d}, is one of the Cartesian coor-dinates of x.

A. Bulk statistics: d−dimensional Levydistributions

We use Eq. (13) to rewrite Eq. (12) in the one-dimensional form,

P (k, u)=

⟨1− ψ(u− ikvx)

u− ikvx

⟩vx

1

1− 〈ψ(u− ikvx)〉vx,(24)

where the averaging is taken over the distribution of vxgiven by Eqs. (15), (18). Thus we can directly use theresults for P (k, t) in the one-dimensional case. The dif-ference from the one-dimensional case is in how the realspace PDF is reproduced from P (k, t): here the formulafor the inverse Fourier transform of radially symmetricfunction in d dimensions has to be used. We find from[17] that the bulk of the PDF is described with,

Pcen(k, t) ∼ exp [−Kαt|k|α] , (25)

Kα =A

〈τ〉〈|vx|α〉

∣∣∣cos(πα

2

)∣∣∣ , (26)

6

FIG. 2: Scaling limits for a three-dimensional uniform Levy walk. a) Levy scaling of the bulk of numerically sampledPDFs P (x, t) = 〈δ (|x(t)| − x)〉. Thin black line corresponds to the PDFs P (x, t) = 〈δ (|x(t)| − x)〉L3

obtain by averaging over

the three-dimensional Levy distribution, Eq. (32); b) Ballistic scaling of the tails of the PDFs. Thin black line is Eq. (39). Both

PDFs were sampled over 1012 realizations. The parameters are α = 3/2, v0 = 1. In our simulation we used ψ(τ) = (3/2)τ−5/2

for τ > 1 otherwise it is zero.

where 〈|vx|α〉 is given by Eq. (20) with γ = α. Here thesubscript in Pcen was introduced in [17]. It stands for thecentre (or bulk) part of the PDF. Briefly, to obtain thisresult we expand P (k, u) using the scaling assumptionthat kα is of the order u when both are small. Later,we will derive these results as a special case of the moregeneral non-isotropic model (see eq. (52) below).

We can write Kα in terms of 〈|v|α〉 using Eq. (22),

Kα =AΓ[(α+ 1)/2]Γ(d/2)

〈τ〉Γ[(d+ α)/2]√π〈|v|α〉

∣∣∣cos(πα

2

)∣∣∣ . (27)

This coefficient reduces to the diffusion coefficient of one-dimensional walk found in [17] setting d = 1. In the casewhere v is a conserved constant v0 (modelling conserva-tion of energy), we find

Kα =AΓ[(α+ 1)/2]Γ(d/2)

〈τ〉Γ[(d+ α)/2]√πvα0

∣∣∣cos(πα

2

)∣∣∣ . (28)

Using the inverse Fourier transform we find for the PDF’s

bulk,

Pcen(x, t) ∼ 1

(Kαt)d/αLd

(x

(Kαt)1/α

), (29)

Ld(x) =

∫exp[ik · x− kα]

dk

(2π)d. (30)

These formulas provide generalized CLT ford−dimensional isotropic LWs. Below these will begeneralized to the case of arbitrary (not necessarilyisotropic) statistics of velocity. We provide the form ofLd(x) in the cases of physical interest d = 2 and d = 3.In the two-dimensional case we have,

L2(x) =

∫ ∞0

kJ0(kx) exp [−kα]dk

2π, (31)

that can be called two-dimensional isotropic Levy den-sity. Here J0(z) is the Bessel function of the firstkind. In three dimensions we find (L3(x) is normalized∫L3(x)dx = 1),

L3(x)=− 1

x(∂x)

∫ ∞0

dk

2π2cos(kx) exp[−kα]=

L′ (x)

2πx, (32)

7

FIG. 3: Probability density functions of Levy walks in three-dimensions. The distributions for the time t/τ0 = 104

were obtained by sampling over 1011 realizations. They are represented by the set of two-dimensional ’slices’ along z-axis,P (x, y, z = const), plotted on a log scale. The other parameters are as in Fig. 1.

where L(x) = L1(x) is the standard Levy distribution,

L(x) =

∫exp (ikx− |k|α)

dk

2π. (33)

Thus in three dimensions the isotropic Levy density canbe obtained from the one-dimensional one by differenti-ation (this is improved version of the old result of [12]valid for arbitrary velocity distribution). This is true forany odd-dimensional case. It can be demonstrated thatin the case of even dimension L2n(x) can be obtainedfrom L(x) using derivative operator of half integer order,see Appendix E and cf. [21].

We find from Eq. (32) that L3(x) ∝ |x|−α−3 at largeargument where we use the well-known behavior L(x) ∼|x|−α−1, see e. g. [14]. (Similarly it will be demon-strated below that Ld(x) ∼ |x|−α−d.) This tail must failat larger arguments because it would give divergent dis-persion 〈x2(t)〉 ∝

∫x2Ld(x)dx =∞, which is wrong pro-

vided that the moments of F (v) are finite, an assumptionwe use all along this paper. The PDF must necessarilydecay fast at x > vtt where vt is the typical value ofvelocity. Thus the Levy density does not provide validdescription of the tail of the PDF that determines dis-persion. This necessitates the study of the tail of thedistribution performed below.

B. Infinite density

The description of the tail of the PDF is performedusing the reduction to one-dimensional case where theproblem was solved in [17]. This solution is based onasymptotic resummation of the series for the character-istic function.

We observe that for isotropic statistics of velocityP (k, t) obeys,

P (k, t) = 〈exp [−ik · x]〉 = 1 +

∞∑n=1

(−1)nk2n⟨x2n1⟩

(2n)!,(34)

where we used that odd moments of k · x vanish and

that isotropy implies that⟨

(k · x)2n⟩

is independent of

direction of k so it can be obtained setting k = kx giving⟨(k · x)

2n⟩

= k2n〈x2n1 〉, cf. with similar consideration

for velocity.Though

⟨x2n1⟩

cannot be found completely at all times,it was discovered in [17] that this can be done asymptot-ically in the limit of large times. The proper adaptationof the result tells that using small k and u expansion ofthe quasi-one-dimensional Montroll-Weiss equation (12)when keeping the ratio k/u fixed we find,

P (k, t)∼1+A

〈τ〉

∞∑n=1

Γ(2n−α)(−1)nt2n+1−α⟨v2nx⟩k2n

(2n−1)!|Γ(1−α)|Γ(2n+2−α).(35)

This result was obtained in the limit of long times asymp-totically that is the n−th term in the series is valid pro-vided time is large. How large this time is depends onn: the higher n is, the larger times are needed for thevalidity of the asymptotic form. Thus at however largebut finite t the terms of the series fail starting from somelarge but finite n. Thus, in contrast, to Eq. (34) the re-summation of the series given by Eq. (35) does not haveto lead to P (x, t) because there is no time for which allterms in the series of Eq. (35) are valid.

It is the finding of [17] that resummation of the se-ries in Eq. (35) still produces function that has physicalmeaning. That function called the infinite density givesvalid description of the tail of P (x, t) but fails in the bulk.

8

This is because the bulk corresponds to small x and largek ∝ 1/x. For very small x very large k are relevant im-plying that terms with very large n become relevant forthe sum. However these terms are not valid at finite tleaving the small x inaccessible for the sum in Eq. (35).

Since statistics of both x and v are isotropic then,

〈x2n1 〉〈x2n〉

=〈v2nx 〉〈v2n〉

, (36)

see Eq. (23). We find comparing the series in Eqs. (34)-(35),

⟨x2n(t)

⟩=

2nA⟨v2n⟩

〈τ〉|Γ(1− α)|(2n+1−α)(2n−α)t2n+1−α.(37)

Remarkably this is independent of dimension and thuscoincides with the one-dimensional case (isotropy im-plies that the geometry disappears from 〈x2n〉 becauseof Eq. (36)). The use of these moments for formal re-construction of the long-time limit of the PDF P (x, t) =〈δ (|x(t)| − x)〉 of |x(t)| through PA(x, t) defined by,

PA(x, t)=

∫dk

2πexp[−ikx]

[1+

∞∑n=1

(ik)2n⟨x2n(t)

⟩(2n)!

], (38)

with⟨x2n(t)

⟩given by Eq. (37) gives on resummation

[17],

PA(x, t) =ASd−1

〈τ〉|Γ(1− α)|tα

∫ ∞|x|/t

vd−1F (v)dv

×[α|v|α

|x/t|1+α− (α− 1)

|v|α−1

|x/t|α

], (39)

that holds for x 6= 0. The function PA(x, t) clearly de-scribes the long-time behavior of the moments of |x(t)|via

〈x2n(t)〉 ∼∫ ∞0

PA(x, t)x2ndx, n ≥ 1. (40)

This function however does not describe the normaliza-tion (obtained as n = 0) since PA(x, t) ∼ x−(1+α) forx → 0. Hence PA(x, t) is not normalizable for whichreason it is called infinite density. However it does de-scribe the integer order moments

∫∞0P (x, t)x2ndx where

P (x, t) is the PDF of the distance to the origin |x(t)|.Thus P (x, t) ∼ PA(x, t) is true for integrals with integerpowers. It can be seen that this function describes thetail of the PDF P (x, t) at x ∼ t (that is at large timesP (x, t) ∼ PA(x, t) holds for large x ∝ t) while the bulkcorresponds to x ∼ t1/α. This fits that the momentsof integer order are determined by the tail of P (x, t) asclarified in the coming Sections. Below we derive the infi-nite density in d dimensions in different form proving theexistence of finite long-time limit limt→∞ td−1+αP (tx, t)for arbitrary (anisotropic) statistics of velocity. The de-scriptions of the bulk and the tail of the PDF togetherwith the moments provide a complete description of thed−dimensional walk with isotropic statistics.

IV. CLT FOR ANISOTROPIC LEVY WALKS

We start the analysis of the anisotropic LWs withderivation of the generalized CLT that describes PDFP (x, t) of the walker. We consider in this Section ran-dom walks whose single step duration’s PDF ψ(u) obeysEq. (6) at small u but we let the range of considered αinclude α = 2. That is we consider ψ(u) in Eq. (6) with1 < α ≤ 2. In the case of α = 2 though ψ(τ) does notobey Eq. (4) with α = 2. This is because Eq. (6) withα = 2 describes the case of finite dispersion of τ given by〈τ2〉 = ψ′′(u = 0) = 2A. In contrast, for α = 2 Eq. (4)gives τ−3 tail for which the dispersion is infinite. Thusψ(τ) obeying Eq. (6) with α = 2 decays faster than τ−3.

We demonstrate that for ψ(u) obeying Eq. (6) with1 < α ≤ 2 there is finite limit,

limt→∞

td/αP (t1/αx, t)

=

∫exp

[ik · x− A

〈τ〉

∣∣∣cos(πα

2

)∣∣∣ 〈|k · v|α〉] dk

(2π)d, (41)

that holds for arbitrary statistics of v obeying F (v) =F (−v). It is proper to call this result generalized CLTbecause for α = 2 it reproduces the central limit theorem,

limt→∞

td/2P (x√t, t)=

∫exp

[ik · x−Γplkpkl

2

]dk

(2π)d, (42)

where the RHS defines g(x) in Eq. (1), the covariancematrix Γ is defined by,

Γpl =〈τ2〉〈vpvl〉〈τ〉

, (43)

and we used A = 〈τ2〉/2. In Eq. (42) we use Einsteinsummation rule over the repeated indices. We observethat the units of k in Eq. (42) are t1/2/l, units of x arel/t1/2 and units of Γ are l2/t so that the argument of theexponent is dimensionless.

The form of the covariance matrix could be seenconsidering the second moment of displacement x(t) =∑N(t)k=1 vkτk,

limt→∞

〈xp(t)xl(t)〉t

= limt→∞

〈N(t)vpvlτ2〉

t=〈τ2〉〈vpvl〉〈τ〉

,

where we used that the law of large numbers implieslimt→∞〈N(t)〉/t = 1/〈τ〉.

In the Gaussian α = 2 case the details of statisticsof individual steps of the walk become irrelevant in thelong-time limit: they get summarized in d(d+ 1)/2 inde-pendent coefficients of the covariance matrix Γ. The sec-ond moments of velocity 〈vpvl〉 determine the long-timestatistics of the displacement uniquely. In contrast inα < 2 case the details of the walk influence the displace-ment’s PDF however large time is via 〈|k · v|α〉. Thisis non-trivial function of direction of k that depends onwhich directions of motion are more probable in one step.

9

This is a function of continuous variable rather than Γthat depends on finite number of discrete indices. Herewe assume that isotropy is broken - in the isotropic casethe degree of universality of α < 2 and α = 2 is thesame: 〈|v|α〉 determines uniquely the long-time behaviorgiven by Eq. (27). Correspondingly infinite variability ofshapes of P (x, t) is possible in contrast with fixed Gaus-sian shape in α = 2 case.

We start the derivation of Eq. (41). The calculationbelow hold for 1 < α ≤ 2. We use

td/αP (t1/αx, t) = td/α∫

dk

(2π)dexp

[it1/αk · x

]P (k, t)

=

∫dk

(2π)dexp [ik · x]P (t−1/αk, t). (44)

Thus we have to prove the existence of the limit,

limt→∞

P (t−1/αk, t)=

∫du

2πiexp[u] lim

t→∞

1

tP

(k

t1/α,u

t

).(45)

We use that (〈k · v〉 = 0),⟨1− ψ(ut−1 − ik · vt−1/α)

ut−1 − ik · vt−1/α

⟩= 〈τ〉+ o(t), (46)

1−⟨ψ

(u

t− ik · vt1/α

)⟩=〈τ〉ut−A

⟨(u

t− ik · vt1/α

)α⟩(47)

+o(t) =〈τ〉u+A| cos(πα/2)| 〈|k · v|α〉

t, (48)

where we used that F (v) = F (−v) implies [17],

〈(−ik · v)α〉 =

⟨|k · v|α exp

[− iπα

2sign (k · v)

]⟩= 〈|k · v|α〉 cos

(πα2

). (49)

This formula holds in α = 2 case as well. We find usingEqs. (46)-(48) in the Montroll-Weiss equation that,

limt→∞

1

tP

(k

t1/α,u

t

)=

1

u+ A| cos(πα/2)| 〈|k · v|α〉,

where A = A/〈τ〉, cf. one-dimensional case in [17]. Weconclude that,

limt→∞

P (t−1/αk, t)=

∫du

2πi

exp[u]

u+ A| cos(πα/2)| 〈|k · v|α〉,(50)

see Eq. (45). We find performing the integration,

limt→∞

P (t−1/αk, t)= exp

[− A

〈τ〉

∣∣∣cos(πα

2

)∣∣∣ 〈|k · v|α〉] ,(51)

completing the proof of Eq. (41). We find asymptoticallyat large times that,

P (k, t) ∼ exp

[− tA〈τ〉

∣∣∣cos(πα

2

)∣∣∣ 〈|k · v|α〉] . (52)

This is one of our main results as it provides the gen-eralized CLT for non-isotropic LWs. For isotropic caseEq. (52) reduces to Eqs. (26).

We characterize different statistics of velocity with

structure function s(k) that depends on the unit vector

k = k/k,

s(k) =Γ[(d+ α)/2]

√π

Γ[(α+ 1)/2]Γ(d/2)

⟨|k · v|α

⟩〈|v|α〉

. (53)

This is defined so that for isotropic statistics s(k) = 1

(in that case we have⟨|k · v|α

⟩= 〈|vx|α〉 where 〈|vx|α〉

is determined by Eq. (22)). We have with this definition,

limt→∞

td/αP (t1/αx, t)=

∫exp

[ik·x−Kαk

αs(k)] dk

(2π)d,

P (k, t) ∼ exp[−Kαtk

αs(k)], (54)

where we use Kα defined in Eq. (27) (in anisotropic caseKα does not have direct interpretation of diffusion coef-ficient so this is to be taken as mathematical definition).The PDF of the displacement obeys,

P (x, t) ∼ 1

(Kαt)d/αLd

(x

(Kαt)1/α

), (55)

Ld(x)=

∫exp

[ik · x− kαs(k)

] dk

(2π)d. (56)

For isotropic model there is no modulation and Ld(x) isthe universal function Ld(x) introduced previously, seeEq. (30). Thus different isotropic statistics producesthe same long-time PDF in the bulk that differ onlyby the value of Kα. For XY Z... model 〈|k · v|α〉 =

vα0∑di=1 |ki|α/d we find that the distribution factorizes

in the product of one-dimensional distributions. Thus inthis case the bulk of the PDF coincides with that of in-dependent walks along different axes. In these cases thefunctional shape is universal.

The chief feature introduced by the passage fromone dimension to the higher-dimensional case is thatquite arbitrary angular structure of the distribution be-

comes possible. The structure function s(k) describespositive angular modulation in k−space that changescorrespondingly the functional form in real space, seeEq. (56). This function does not seem to obey strongconstraints that would strongly limit the possible formsof Ld(x). We stress this fact calling distribution Ld(x)’anisotropic d−dimensional Levy distribution’ in contrastwith d−dimensional isotropic Levy distributions intro-duced previously [12] in the context of Levy flights thathave universal shape.

Considering marginal distributions of components ofx one reduces the problem to one dimension restoringuniversality of the distribution. Integrating Eq. (55),

P (xi, t)=

∫P (x, t)

∏k 6=i

dxk∼1

(Kit)1/αL

(xi

(Kit)1/α

),(57)

10

where L(x) is defined in Eq. (33) and Ki is ”diffusioncoefficient in i−th direction”,

Ki =A

〈τ〉

∣∣∣cos(πα

2

)∣∣∣ 〈|vi|α〉 . (58)

Thus marginal PDFs are given by the standard one-dimensional symmetric Levy stable law. In contrast thePDF P (x, t) =

∫δ(|x(t)| − x)P (x, t)dx of the distance

x(t) = |x(t)| from the origin is not universal. We findintegrating Eq. (55) that,

P (x, t) ∼ 1

t1/αP0

( x

t1/α

), (59)

P0(x)=

∫xd/2Jd/2−1(kx)dk

(2π)d/2kd/2−1exp

[−Kαk

αs(k)], (60)

(see further details for notation choices in the next Sec-tion) where we used∫

exp [ik · x] δ(|x(t)| − x)dx =(2πx)d/2Jd/2−1(kx)

kd/2−1,

where Jν(z) is the Bessel function of the first kind of orderν. We observe that integration over angles implied bythe definition of the PDF of the distance from the origindoes not bring universal form of the PDF (that wouldthen coincide with the PDF for isotropic statistics). Thestructure function is present in Eq. (60) and can producequite different forms of P (x, t). We have using Taylorseries for the Bessel function in Eq. (60),

P0(x) = xd−1∞∑n=0

(−1)ncnn!Γ(d/2 + n)2d/2−1

(x2

)2n, (61)

cn =

∫dk

(2π)d/2k2n exp

[−Kαk

αs(k)],

where xd−1 factor describes the contribution of the sur-face of the sphere of radius x. In the isotropic case wehave,

cn=

∫dk

(2π)d/2k2n exp [−Kαk

α]=Sd−1Γ[(2n+d)/α]

(2π)d/2αK(2n+d)/αα

.

We find using that in isotropic case P0(x) =P0(x)/[xd−1Sd−1] the Taylor series for Pcen(x, t) definedin Eq. (30),

Ld(x) =

∞∑n=0

(−1)nΓ[(2n+d)/α]

n!Γ(n+ d/2)22n+d−1πd/2αx2n, (62)

The factor of Γ(n + d/2) can be simplified in cases ofodd and even dimensions using Γ(n+ 1) = n! and Γ(n+1/2) = 2−n

√π(2n − 1)!!. In the case of d = 1 the series

reproduces the one-dimensional formula [14].As mentioned, for arbitrary statistics of velocity there

seems to be no constraint on the Taylor coefficients cnthat would determine uniquely the functional form ofP0(x). However despite that the small x expansion ofP0(x) is not universal, the large x behavior of P0(x) isuniversal. This will be demonstrated in the next Section.

V. UNIVERSAL TAIL OF ANISOTROPIC LEVYDISTRIBUTION AND LOW-ORDER MOMENTS

In this Section we demonstrate that the PDF of thedistance of the walker from the walk’s origin has power-law tail with universal (independent of statistics of veloc-ity) exponent. This has the implication that moments ofdistance from the origin with order smaller than α are de-termined by the bulk of the PDF but moments of higherorder are determined by the PDF’s tail (whose descrip-tion is different from Levy distribution and is providedfurther in the text). Thus dispersion is determined by thetail of the PDF independently of the statistics of velocity.

We observe from Eq. (60) that,

P0(x)− δ(x) = xd/2∫

dk

(2π)d/2k1−d/2Jd/2−1(kx)

×(

exp

[− A〈τ〉

∣∣∣cos(πα

2

)∣∣∣ 〈|k · v|α〉]− 1

),

where we used the Fourier transform representation ofthe δ−function and restored the definitions of the con-stants. By using the large argument asymptotic expan-sion of Bessel function and rescaling integration variableby x, we find

P0(x)∼√

2

x√π

∫dk

(2π)d/2k(1−d)/2 cos

(k − π(d− 1)

4

)×(

exp

[−A 〈|k · v|

α〉xα|〈τ〉

∣∣∣cos(πα

2

)∣∣∣]− 1

)exp [−εk] ,(63)

where infinitesimal ε in the last term is the convergencefactor introduced for convergence of the large x expan-sion found by expanding the exponent in brackets. Theleading order term is,

P0(x)∼− A√

2

x1+α√π〈τ〉

∣∣∣cos(πα

2

)∣∣∣ ∫ k(1−d)/2dk(2π)d/2

×〈|k · v|α〉 cos

(k − π(d− 1)

4

)exp [−εk] . (64)

The angular integral gives∫dk⟨|k · v|α

⟩= 〈vα〉

∫dk|k · v|α, (65)

where we interchanged the orders of averaging and inte-gration and used that the last integral is independent ofv. However since it is independent of v then it can beobtained taking v in x−direction which gives,∫

dk⟨|k · v|α

⟩= 〈vα〉

∫dk|kx|α, (66)

We find using this in Eq. (64),

P0(x)∼−√

2Γ[(d+ α)/2]Kα

x1+αΓ[(α+ 1)/2]Γ(d/2)

∫k(1−d)/2dk

(2π)d/2

×|kx|α cos

(k − π(d− 1)

4

)exp [−εk] , (67)

11

where we use Kα defined in Eq. (27). It is seen read-ily from Eq. (63) that the condition of applicability of

large x expansion is x� K1/αα (cf. Eq. (59) for checking

the units). The x−axis defining kx in the integrand isarbitrary direction in space. We observe that |kx|α aver-aged over the angles can be found using in Eq. (23) theisotropic statistics with x switched by k:

〈|kx|α〉angle =Γ[(α+ 1)/2]Γ(d/2)

Γ[(d+ α)/2]√π|k|α, (68)

where we set 〈|k|α〉 = |k|α. We find,

P0(x)∼−√

2KαSd−1x1+α(2π)d/2

√π

∫ ∞0

kα+(d−1)/2dk

× cos

(k − π(d− 1)

4

)exp [−εk] ,

where we performed the integral over angles. We write,

P0(x) ∼ −√

2KαSd−1x1+α(2π)d/2

√π

Re

[exp

(iπ(d− 1)

4

)×∫ ∞0

exp [−(ε+ i)k] kα+(d−1)/2dk

]. (69)

By using integration variable t = (ε+ i)x we find

P0(x) ∼ −√

2KαSd−1Γ(α+ (d+ 1)/2)

x1+α(2π)d/2√π

Re

[exp

(iπ(d− 1)

4

)(ε+ i)−α−(d+1)/2

]. (70)

We conclude that at x� K1/αα ,

P0(x) ∼√

2KαSd−1Γ(α+ (d+ 1)/2) sin(πα/2)

x1+α(2π)d/2√π

. (71)

For the probability density function we find restoringphysical units that at x� (Kαt)

1/α,

P (x, t) ∼√

2KαSd−1Γ(α+ (d+ 1)/2) sin(πα/2)t

x1+α(2π)d/2√π

, (72)

where we used Eq. (59). Other form is obtained usingthe value of Kα given by Eq. (27),

P (x, t)∼ Γ[(α+ 1)/2]Γ(α+ (d+ 1)/2)tA〈|v|α〉2(d−1)/2|Γ(1−α)|Γ(α)Γ[(d+α)/2]〈τ〉x1+α

,(73)

where we used Γ(α)Γ(1 − α) = π/ sin(πα). In three-dimensional and one-dimensional cases we find,

P (x, t) ∼ αtA〈|v|α〉|Γ(1− α)|〈τ〉x1+α

, d = 1, 3. (74)

In one dimension this is known result [14, 17, 18]. Thecoincidence of tails in one- and three-dimensional casesdoes not have clear origin. In other dimensions including

the physically relevant tow-dimensional case the tail isdifferent and dimension-dependent.

We conclude that despite that the PDF of x is non-universal, its tail obeys universal power-law with decayexponent α+ 1. Furthermore though the velocity statis-tics can be non-isotropic the tail is determined uniquelyby 〈vα〉 so effective isotropization occurs. This could lookcontradicting the non-isotropy of both the bulk and thetail of the PDF demonstrated in previous and comingSections. In fact isotropization happens only in the lead-ing order term of large x series of P (x, t). It is seen read-ily from Eq. (63) that the next order term involves the

angular integral∫dk⟨|k · v|α

⟩2that depends on non-

isotropy of the velocity statistics (the orders of integra-tion and averaging can no longer be interchanged becauseof non-linearity). Thus the large x series of P (x, t) isisotropic in leading order only.

We illustrate the conclusions of this Section with two-dimensional case of XY Z... model, called non-symmetricx− y model. In this model diffusion coefficients in x andy directions differ so diffusion is non-symmetric. Thevelocity vector is in x or −x direction with probabilityλ/2 and in y or −y directions with probability (1−λ)/2.The magnitude of velocity v0 is fixed. Thus 〈|k · v|α〉 =vα0 [λ|kx|α + (1− λ)|ky|α]. We find from Eq. (41) that,

P (x, t) ∼ 1

(KxKyt2)1/αL

(x

(Kxt)1/α

)L

(y

(Kyt)1/α

),

Kx=Avα0 λ

〈τ〉

∣∣∣cos(πα

2

)∣∣∣ , Ky=Avα0 (1−λ)

〈τ〉

∣∣∣cos(πα

2

)∣∣∣ .This distribution is non-symmetric where asymmetry re-sults from different scaling factors of one-dimensional dis-tributions in x and y−directions. The PDF P (x, t) of thedistance x from the origin is (in the previous equation xis x−component of x, below x = |x| with no ambiguity),

P (x, t) ∼ x

(KxKyt2)1/α

∫ 2π

0

L

(x cosφ

(Kxt)1/α

)×L

(x sinφ

(Kyt)1/α

)dφ. (75)

This PDF depends on the degree of asymmetry via λincluded in Kx, Ky. In the limits of λ → 0 or λ → 1it gives LW in the direction of the corresponding axis.When λ = 1/2 we find the distribution of x−y model. Incontrast the large x asymptotic form of P (x, t) dependson 〈|v|α〉 = vα0 only and thus is independent of λ, seeEq. (73). The simplest way of verifying this seems to beusing integral representation of L(x) and repeating thesteps of the previous derivation in the general case.

The universal power-law tail obtained above has non-trivial implications for the moments of distance from theorigin. The moment of q−th order 〈xq(t)〉 diverges atlarge x is q > α. Consequently the second moment isdetermined by the tail of the PDF independently of thestatistics of velocity. The moments of order 0 < q < α

12

are determined by the bulk of the PDF. We presumethat the moments are determined either by the bulk orby the tail of the PDF which is usually the case and isconfirmed below. Then demanding self-consistency wefind the described conclusions. We find from Eqs. (59),

〈|x|q(t)〉 ∼ cqtq/α, q < α (76)

where the constants cq are given by

cq =

∫ ∞0

xqP0(x)dx, (77)

where P0(x) is given by Eq. (60). For q > α the integraldiverges and the formula fails demanding finding the tailof the PDF. This is obtained in the next Section.

VI. INFINITE DENSITY FOR ANISOTROPICLEVY WALKS

In this Section we demonstrate that there is the finitelimit,

limt→∞

td−1+αP (tw, t), w 6= 0. (78)

In the next Section we demonstrate that this limit pro-vides the description of the tail of the PDF P (x, t) onscales where |x| ∝ t.

We introduce P (x, t) = P (x, t) − δ(x). We havefrom the Montroll-Weiss equation (12) that the Fourier-

Laplace transform P (k, u) of P (x, t) obeys,

P (k, u)=

⟨1−ψ(u−ik · v)

u−ik · v

⟩1

1−〈ψ(u−ik · v)〉− 1

u.(79)

We use,

P (tw, t) =

∫dk′

(2π)ddu′

2πiexp [itk′ ·w + u′t] P (k′, u′)

= t−d−1∫

dk

(2π)ddu

2πiexp [ik ·w] P

(k

t,u

t

). (80)

We prove the existence of finite limit,

Pi(k, u) = limt→∞

tα−2P

(k

t,u

t

), (81)

where i stands for ”infinite”. We have

limt→∞

P (tw, t)

t1−d−α=

∫dk

(2π)ddu

2πiexp [ik ·w+u] Pi(k, u).(82)

We use the large t asymptotic forms (〈k · v〉 = 0) for ψwhich is Laplace transform of ψ(τ),⟨1−ψ(ut−1−ik · vt−1)

ut−1−ik · vt−1

⟩∼〈τ〉−A

⟨(u

t− ik · v

t

)α−1⟩,

and [1−⟨ψ

(u

t− ik · v

t

)⟩]−1∼ t

〈τ〉u(83)

+(ut

)α−2 A

〈τ〉2

⟨(1− ik · v

u

)α⟩, (84)

holding when the rest of the arguments are held fixed.We neglected higher order terms.

The use of these identities in Eq. (79) gives,

u2−αPi(k, u)

A/〈τ〉=

⟨(1− ik · v

u

)α⟩−

⟨(1− ik · v

u

)α−1⟩.

We find using Eq. (82) and P (x, t) = P (x, t) for x 6= 0,

limt→∞

P (tw, t)

t1−d−α=

A

〈τ〉

∫dk

(2π)ddu

2πiuα−2 exp [ik ·w + u][⟨(

1− ik · vu

)α⟩−

⟨(1− ik · v

u

)α−1⟩], w 6= 0. (85)

It can be seen readily that in one dimension Eq. (85) re-produces the infinite density obtained in [17]. We can usethe way of calculation proposed in that work for findingthe integral in Eq. (85). We use series,

(1− x)α − (1− x)α−1 =

∞∑n=1

(−α)nxn

α(n− 1)!(86)

where (a)n = Γ(a+n)/Γ(a) = a(a+ 1)..(a+n−1) is thePochhammer symbol. We find,

limt→∞

P (tw, t)

t1−d−α=A

〈τ〉

∫dk

(2π)ddu

2πiuα−2−2n exp [ik·w+u]

(−α)2n⟨[ik·v]2n

⟩α(2n−1)!

=A

〈τ〉

∫dk

(2π)dexp [ik·w] 〈Gα(k·v)〉

|Γ(1− α)|,(87)

where we defined,

Gα(y) =

∞∑n=1

(−1)ny2n

(2n− 1)!(2n− α)(2n+ 1− α), (88)

using (−α)2n/Γ(2n+2−α) = (2n−α)(2n+1−α)/Γ(−α).The function Gα(y) was introduced in [17] where it wasdemonstrated that,

G(y) = αBα(y)− (α− 1)Bα−1(y), (89)

Bα(y) =

∫ 1

0

cos(ωy)− 1

ω1+αdω. (90)

Thus for finding the inverse Fourier transform (87) weconsider,

Bα(w,v) =

∫dk

(2π)dexp [ik·w] Bα(k · v)

=

∫dk

(2π)dexp [ik·w]

∫ 1

0

cos(ωk · v)− 1

ω1+αdω. (91)

13

We observe that,∫dk

(2π)dexp [ik·w] cos(ωk · v)=

δ(w+ωv)+δ(w−ωv)

2.

Thus we find,

Bα(w,v)=

∫ 1

0

δ(w+ωvv)+δ(w−ωvv)− 2δ(x)

2ω1+αdω.(92)

where v = vv. We find using the angular δ−functionδa(n) obeying δ(w − w′) = w1−dδ(w − w′)δa(w − w′)that for w 6= 0,

Bα(w,v)=vα∫ v

0

δ(w−ω′)δa(w+v)+δa(w−v)

2ω′1+αwd−1dω′,(93)

where ω′ = vω. This gives that,

Bα(w,v) =δa(w + v) + δa(w − v)

2v−αwd+α, |w| < |v|, (94)

Bα(w,v) = 0, |w| > |v|. (95)

We find using Eqs. (87),(89) that (we switch w with di-mensions of velocity by v in the final formula) for v 6= 0,

I(v) = limt→∞

P (tv, t)

t1−d−α=

A

vd−1|Γ(1− α)|〈τ〉

×∫v′>v

F (v′v)v′d−1dv′[αv′α

v1+α− (α− 1)

v′α−1

vα

], (96)

where I(v) is the infinite density. We separated the de-pendence in the velocity’s PDF F (v) on the magnitudev and direction v and used F (v) = F (−v). This is chiefresult of our work that provides the infinite density as thestatement on the existence of long-time scaling limit ofthe PDF similar to the central limit theorem or the limitintroduced in the previous Section. The point v = 0describes the region of not too large |x| where Levy dis-tribution describes P (x, t) well. This region that deter-mines the normalization shrinks in the considered largetimes’ scaling limit to the point x = 0.

Infinite density inherits anisotropy of F (v): all angularharmonics present in the expansion of F (v) in sphericalharmonics will be present in the expansion of I(v), seeEq. (96) (disregarding degenerate cases when the inte-gral that provides the corresponding coefficient vanishes).Thus in contrast with the case of isotropic statistics de-scribed by Eq. (39) the angular structure is non-isotropicwhen the velocity statistics is not. Furthermore, thoughanisotropy of the statistics of the single step of the walkinfluences the distribution in the bulk, described by theanisotropic Levy distribution, for the tail this influenceis more immediate.

The infinite density limit (96) implies that at largetimes,

P (x, t) ∼ 1

td+α−1I(xt

), x 6= 0. (97)

This is complementary to the other limit implied byEq. (54) ,

PL(x, t) ∼ 1

(Kαt)d/αLd

(x

(Kαt)1/α

). (98)

In fact, both limits characterize different asymptotic re-gions of the PDF. It is clear from the structure of thescaling limits that Levy distribution describes the bulkof the PDF whose scale grows proportionally to t1/α andinfinite density describes the tail with |x| ∝ t. In thenext Section we demonstrate that the integer order mo-ments are determined by the infinite density tail of thePDF.

VII. INTEGER ORDER MOMENTS ANDDISPERSION AT LARGE TIMES

In this Section we calculate the long-time limit of themoments of integer order from the Montroll-Weiss equa-tion. The basic result that we derive is that

〈xi(t)xk(t)〉 =2At3−α

|Γ(1−α)|(2−α)(3−α)〈τ〉〈vivk〉. (99)

The property F (v) = F (−v) implies that 〈vivk〉 = 0 fori 6= k so we can write,

〈xi(t)xk(t)〉 =2At3−α

|Γ(1−α)|(2−α)(3−α)〈τ〉〈v2i 〉δik. (100)

The trace of this equation gives,

〈x2(t)〉 =2A〈v2〉

|Γ(1−α)|(2−α)(3−α)〈τ〉t3−α. (101)

This is universal formula that holds in arbitrary dimen-sion for arbitrary (possibly strongly anisotropic) statis-tics of velocity. In the case of isotropic statistics thisreduces to the previously derived Eq. (37). In one di-mension this reproduces the formula of [17].

We observe that directly repeating the steps of one-dimensional calculation in [17] with k ·v instead of kv wefind the asymptotic expansion,

P (k, t)∼1+A

〈τ〉

∞∑n=1

Γ(2n−α)(−1)nt2n+1−α⟨(k · v)2n

⟩(2n−1)!|Γ(1−α)|Γ(2n+2−α)

.

which is direct continuation of Eq. (36) for arbitrarystatistics of velocity. For XY Z... model we have⟨(k · v)2n

⟩= v2n0 [

∑di=1 k

2ni ]/d so that P (k, t) has the

structure of P (k, t) =∑di=1 P1(ki, t) where P1(k, t) is

the corresponding one-dimensional distribution. Thusthe inverse Fourier transform is sum of products on one-dimensional distribution P1(xi, t) times δ−functions ofthe rest of coordinates. For instance in three dimensionswe have

P (x, t) ∼ P1(x, t)δ(y)δ(z) + δ(x)P1(y, t)δ(z)

+δ(x)δ(y)P1(z, t),

14

where P1(x, t) is the one-dimensional distribution stud-ied in [17]. Correspondingly the temporal growth of themoments is as in one dimension.

Comparing the asymptotic form of P (k, t) with thecharacteristic function, see Eq. (34),

P (k, t) = 1 +

∞∑n=1

(−1)n⟨(k · x)2n

⟩(2n)!

, (102)

we read off the moments. We find,⟨d∏i=1

xνii (t)

⟩∼

2nA⟨∏d

i=1 vνii

⟩t2n+1−α

|Γ(1− α)|(2n− α)(2n+ 1− α)〈τ〉,(103)

where∑di=1 νi = 2n. The n = 1 term gives Eq. (100). On

the level of dispersion there is no qualitative differencebetween isotropic and anisotropic statistics of velocity.The formula (101) for 〈x2(t)〉 is dimension-independent.This is also true for 2n−th moment of the magnitude ofthe distance from the walk’s origin 〈x2n(t)〉,

〈x2n(t)〉 ∼ 2nA〈v2n〉t2n+1−α

|Γ(1− α)|(2n− α)(2n+ 1− α)〈τ〉,

as can be seen opening brackets in 〈x2n(t)〉 =⟨(∑di=1 x

2i (t)

)n⟩and using Eq. (103). This reproduces

Eq. (37) in the isotropic case and reproduces the resultof [17, 18] for 〈x2n(t)〉 in one dimension. Similarly thegrowth of one of the components is universal:

〈x2ni (t)〉 ∼ 2nA〈v2ni 〉|Γ(1− α)|(2n− α)(2n+ 1− α)〈τ〉

t2n+1−α.

It is readily confirmed using Eq. (103) and the construc-tion of the infinite density in the previous Section thatthe identity⟨

d∏i=1

xνii (t)

⟩∼ 1

td+α−1

∫ d∏i=1

xνii I(xt

)dx,

holds. The calculation can be done by direct transfer ofthe calculation in one dimension [17] using k · v insteadof kv in the derivations. Thus integer order moments aredetermined by the infinite density. We saw previouslythat the moments of order higher than α are describedby the tail of the PDF. This confirms that the tail of thePDF is indeed described by the infinite density (unlessthe tail cannot be reconstructed from moments of integerorder which is not the case here).

Previous results on the moments coincided with thosein one dimension [17]. The difference from the one-dimensional case holds when cross correlations of differ-ent components of x(t) are considered. We consider thedifference in the case of fourth-order moments. We have,

〈x2i (t)x2k(t)〉 ∼ 4A〈v2i v2k〉|Γ(1− α)|(4− α)(5− α)〈τ〉

t5−α.(104)

In XY Z... model 〈v2i v2k〉 = 0 for i 6= k so that this for-mula gives zero. This does not tell that the positive quan-tity x2i (t)x

2k(t) has zero average but rather this tells that

higher order corrections for the asymptotic calculation atlarge times is needed. This can be done by direct studyof the Montroll-Weiss equation.

VIII. TAIL OF THE PDF OF ANISTOROPICLW AND FRACTIONAL ORDER MOMENTS

We observe from Eq. (96) that angular structure ofthe infinite density, and thus of the PDF’s tail, reflectsdirectly the angular structure of the velocity PDF. Thisis the consequence of that the tail is formed by rare longballistic flights. We find for the angular average,

I0(v) =

∫I(vv)dv =

A

vd−1|Γ(1− α)|〈τ〉

∫v′>v

F (v′)dv′

×[αv′α

v1+α− (α− 1)

v′α−1

vα

]. (105)

We conclude from Eq. (97) that the probability densityfunction of the distance x from the walk’s origin obeys,

P (x, t)=xd−1∫P (xx, t)dx∼ xd−1

td+α−1I0

(xt

)=

A

|Γ(1−α)|∫v′>x/t

F (v′)dv′

〈τ〉

[αtv′α

x1+α− (α− 1)

v′α−1

xα

]. (106)

This has universal asymptotic form at small x,

P (x, t) ∼ αtA〈vα〉|Γ(1−α)|〈τ〉x1+α

,tvcx� 1, (107)

where vc is characteristic value of velocity. This is inde-pendent of dimension and details of statistics of velocity.This coincides with the tail of the Levy distribution inone and three-dimensional cases, see Eq. (74). In one-dimension this was observed in [17]. However in otherdimensions there is a multiplicative factor difference be-tween the two asymptotic forms. For I0(x) we have,

I0(x) ∼ αA〈vα〉|Γ(1−α)|〈τ〉xd+α

. (108)

It can be demonstrated that the small x form of the infi-nite density and the large x form of the Levy distributionhave to be of the same order. We observe that PL(x, t)in Eq. (98) obeys at large t and v 6= 0

PL(tv, t)

t1−d−α∝ 1

t1−d−α+d/αLd

(t1−1/α

v

(Kα)1/α

)∝ const,

where we use that at large arguments Ld(x) ∝ x−d−α.Thus PL(x, t) would give finite constant contribution inthe scaling limit described by the infinite density. Thiscontribution comes from the tail of the Levy distribution.Then the consideration performed in one-dimensional

15

case in [17] holds in higher-dimensional case demonstrat-ing that the small x form of the infinite density and thelarge x form of the Levy distribution have to be of thesame order.

Based on the infinite density tail we can readily derivethe moments of order higher than α. We have,

〈|x(t)|q〉 ∼∫|x|q

td+α−1I(xt

)dx = cqt

q+1−α, (109)

cq =

∫|x|q+d−1I0 (x) dx.

This formula holds for q > α where the formula’s break-down at smaller q is signalled by the small x divergenceof the integral in cq see Eq. (108).

The scaling exponents of the moments depend on themoment’s order linearly both at q < α and q > α, albeitwith different linear dependencies. This bilinear behaviorof the moments of the distance from the origin was ob-served in the one-dimensional case in [17] and continuesto hold in the higher-dimensional case, cf. [48].

We find the infinite density in the uniform model withfixed velocity v0. In this case the PDF of |v| is δ(v− v0)so that

I(x)=Avα−10 (α−1)

|Γ(1−α)|τxα

[αv0

x(α−1)−1

], x < v0, (110)

If x > v0 then I(x) = 0. Since xc(t)/t ∝ t(1−α)/α isdecaying function of time then at large times the infinitedensity describes non-trivial region of the the probabilitydensity function. We find that if xc(t) < x < v0t then,

Pd(x, t) ∼AΓ(d/2)vα−10 (α−1)

2πd/2xd+α−1|Γ(1−α)|τ

[αv0t

x(α−1)−1

].(111)

The probability density function at x > v0t vanishes sincethe particle moving at constant speed v0 cannot pass dis-tances longer than v0t in time t.

IX. CONCLUSIONS

We obtained the PDF P (x, t) for d− dimensional Levywalks. We derived two complimentary limiting distribu-tions describing the PDF at long times.

One of these scalings has origin similar to the centrallimit theorem providing the d−dimensional counterpartof one-dimensional Levy distribution that describes thebulk of the PDF. There is however significant differencefrom the one-dimensional case. In one dimension differ-ent statistics of velocity result in the same universal (upto rescaling of the density) shape of the bulk of the PDFP (x, t). This is no longer true in higher dimensions: dif-ferent velocity statistics result in different shapes of thePDF bulks. Only in the case of isotropic statistics a cer-tain universality holds. This is in sharp contrast withthe normal diffusion, where the d−dimensional Gaussianprofiles are universal attractors.

We demonstrated that despite the non-universality ofthe PDF in the bulk region, the tail of PDF of thedistance follows the universal power-law with exponent−1−α. The prefactor of the law depends only on angle-averaged statistics of velocity via 〈|v|α〉 (in contrast thebulk of the PDF depends on probabilities of moving indifferent directions). That provides direct generaliza-tion of the one-dimensional result. Thus dependence onanisotropy of velocity statistics and non-universality ofthe shape of the PDF in the bulk disappear at large dis-tances where universality is restored. The higher ordercorrections are anisotropic and non-universal though.

This universality of the tail’s exponent guarantees uni-versal behavior of the moments of x. The moments oforder smaller than α where 1 < α < 2 are determined bythe PDF’s bulk. However higher order moments, includ-ing dispersion, are due to the PDF’s tail.

The existence of the complimentary scaling, unfamil-iar in the field until very recently, is the consequenceof the scaling of the tail of the PDF ψ(τ) of the singlestep duration τ . This limit defscribes the tail of P (x, t)formed by ballistic motions with very large duration τ . Itdescribes x that scale proportional with t (ballistic scal-ing). In contrast the displacements described by the bulkof the PDF are formed by accumulation of lots of typi-cal diffusive increments. Thus the bulk describes x thathave (anomalous) diffusive scaling proportional to t1/α

(we remark that t1/α � t at large t because of α > 1).

The function that describes the PDF’s tail is calledthe ’infinite density’ because it is not normalizable. Thatis caused by divergence of normalization at small argu-ments. It is important to understand that ’infinite den-sity’ is not a probability density function (which mustbe normalized). The infinite density is a point-like limitof the rescaled PDF and does not have to be normal-izable. This is because for different spatial argumentsthe pointwise limit holds at different times. No matterhow large time is, the infinite density never convergesto the PDF uniformly in space. For any large but finitetime the bulk of the PDF that determines the normaliza-tion holds around the origin x = 0 and can be describedby the corresponding anisotropic Levy distribution. Thecomplete picture of the PDF is simpler in three and one-dimensional cases. There the small argument form ofthe infinite density tail continuously transforms in theuniversal tail of the PDF’s bulk. This makes it plausi-ble that intermediate region between the bulk and thetail is described by these asymptotic forms. Both thebulk anisotropic Levy distribution and its tail shrink tozero when rescaled with ballistic scaling t of the infinitedensity scaling limit. In the two-dimensional case the in-termediate region between the bulk and the tail demandsseparate study.

Our work provides foundation of the sub-ballisticsuper-diffusive Levy walks in a space of arbitrary dimen-sion d.

Acknowledgment. This work was supported by theRussian Science Foundation grant No. 16-12-10496 (SD

16

and VZ). IF and EB thank the Israel Science Foundation for the support.

[1] Ya. G. Sinai, Introduction to Ergodic Theory, (PrincetonUniversity Press, 1976).

[2] L. A. Bunimovich, et al., Hard Ball Systems and theLorentz Gas, (Springer Science and Business Media,2013).

[3] E. Montroll and G. Weiss, J. Math. Phys. 6, 167 (1965).[4] H. Scher and M. Lax, Phys. Rev. B 7, 4491 (1973).[5] M. Shlesinger, J. Stat. Phys. 10, 421 (1974).[6] H. Scher and E. W. Montroll, Phys. Rev. B 12, 2455

(1975).[7] M. F. Shlesinger, J. Klafter, and Y. M. Wong, J. Stat.

Phys. 27, 499 (1982).[8] J. P. Bouchaud and A. Georges, Phys. Rep. 195, 127

(1990).[9] M. B. Isichenko, Rev. Mod. Phys. 64, 9611043(1992).

[10] M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature363, 3137(1993).

[11] J. Klafter, M. F. Shlesinger, and G. Zumofen, Phys. To-day, 33(1996).

[12] V. V. Uchaikin and V. M. Zolotarev, Chance and Stabil-ity: Stable Distributions and their Applications (Walterde Gruyter, 1999).

[13] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000),ISSN 0370-1573.

[14] C. Godreche and J. M. Luck, J. Stat. Phys., 104, 489(2001).

[15] J. Klafter and I. Sokolov, Phys. World 18, 29 (2005).[16] N. Gal and D. Weihs, Phys. Rev. E, 81, 020903 (2010).[17] A. Rebenshtok, S. Denisov, P. Hanggi, and E. Barkai,

Phys. Rev. E 90, 062135 (2014).[18] A. Rebenshtok, S. Denisov, P. Hanggi, and E. Barkai,

Phys. Rev. Lett. 112, 110601 (2014).[19] D. Froemberg, M. Schmiedeberg, E. Barkai, and V.

Zaburdaev, Phys. Rev. E 91, 022131 (2015).[20] V. Zaburdaev, S. Denisov, and J. Klafter, Rev. Mod.

Phys., 87, 483 (2015).[21] M. Magdziarz and T. Zorawik, arxiv, 1510.05614.[22] V. Zaburdaev, I. Fouxon, S. Denisov, and E. Barkai,

arXiv, 1605.02908.[23] J. P. Taylor-King, R. Klages, S. Fedotov, and R. A. Van

Gorder, Phys. Rev. E 94, 012104 (2016).[24] D. A. Kessler and E. Barkai, Phys. Rev. Lett. 108,

230602 (2012).[25] R. Burioni, L. Caniparoli, and A. Vezzani, Phys. Rev. E,

81, 060101 (2010).[26] M. Dentz, T. Le Borgne, D. R. Lester, and F. P. de Bar-

ros, Phys. Rev. E, 92, 032128 (2015).[27] E. Agliari and R. Burioni, Phys. Rev. E, 80, 031125

(2009).[28] The regime with infinite mean step time is distinctive

even in the one-dimensional case; see [17–19, 21].[29] Ch. Bingham, Ann. Stat. 2, 1201 (1974).[30] K. V. Mardia and P. Jupp, Directional Statistics (John

Wiley and Sons, 2000).[31] J. P. Bouchaud and P. Le Doussal, J. Stat. Phys. 41, 225

(1985).[32] G. Cristadoro, Th. Gilbert, M. Lenci, D. P. Sanders,

Phys. Rev. E 90, 050102 (2014).

[33] I. Fouxon, S. Denisov, E. Barkai, to be published.[34] R. S. Ellis, Entropy, Large Deviations, and Statistical

Mechanics, vol. 271, (Springer Science and Business Me-dia, 2012).

[35] R. S. Ellis, Physica D, 133, 106 (1999).[36] E. Balkovsky and A. Fouxon, Phys. Rev. E 60, 4164

(1999).[37] A. Rebenshtok, S. Denisov, P. Hanggi, and E. Barkai,

Math. Model. Nat. Phenom. 11, 76 (2016).[38] C. Godreche and J. M. Luck, J. Stat. Phys. 104, 489

(2001).[39] C. Itzykson and D. Drouffe, Statistical Field Theory:

Volume 1, From Brownian Motion to Renormalizationand Lattice Gauge Theory, (Cambridge University Press,Cambridge, UK, 1989).

[40] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,Series, and Products, (Academic Press, New York, US,2000).

[41] O. Arizmendi and V. Prez-Abreu, Commun. Stoch. Anal4, 161 (2010).

[42] H. Bateman, Higher Transcendantal Functions, vol. 2(New York: McGraw-Hill, 1955).

[43] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theoryand Applications of Fractional Differential Equations,(North - Holland Mathematics Studies 204, Amsterdam,2006).

[44] J. L. Lovoie, T. J. Osler, and R. Tremblay, FractionalDerivatives and Special Functions, SIAM Rev. 18, 240(1976).

[45] M.-F. Li, J.-R. Ren, and T. Zhu, arXiv:0910.4819.[46] G. M. Zaslavsky, Phys. Rep. 371, 461 (2002).[47] W. Feller, An Introduction to Probability Theory, vols. 1

and 2, (Wiley, New York, 1971).[48] P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi,

and A. Vulpiani, Physica D, 134, 75 (1999).[49] P. Levy, Theorie de l’addition des variables aleatoires,

(Gauthiers-Villars, Paris, 1937).[50] J. Aaronson, An Introduction to Infinite Ergodic Theory,

(American Mathematical Society, Providence, RI, 1997).[51] M. Thaler and R. Zweimuller, Probab. Theor. Relat.

Fields 135, 15 (2006).

Appendix A: Montroll-Weiss equation from sumover trajectories

In this Section we derive Montroll-Weiss (MW) equa-tion by using the Laplace transform of the characteristicfunction of the coordinate of random walker. Thoughthis equation is not solvable in general case, it doeshelp in evaluating the long-time asymptotic form of thePDF P (x, t). The MW-equation is well-known in one-dimensional setup, see e. g. [20] and references therein.Here we provide alternative derivation in terms of sumsover all possible trajectories. This can help in the studyof multi-time statistics of the walk, fluctuations, andother quantities.

17

Trajectories during the period [0, t] can be character-ized by the integer number N ≥ 0 of time renewals thatoccurred during this time interval. Since events with dif-ferent N form a complete set of non-overlapping events,we can write averages as sums of contributions of eventswith different N . This is realized by using identity

1 =

∞∑N=1

∫ t

0

dt′∫ ∞t−t′

dt′′δ

(N∑i=1

τi − t′)δ (τN+1 − t′′)

+

∫ ∞t

δ(τ1 − t′)dt′, (A1)

that holds for arbitrary infinite sequence of positive num-bers τi and t. Only one term on the RHS is non-zeroand is one. This is the term of N renewals for which≤∑Ni=1 τi < t <

∑N+1i=1 τi or the last term of no renewals

if τ1 > t. The identity can be used for averaging ar-bitrary function of τi as sum of contributions of eventswith different N . The simplest is the probability PN (t)of having N renewals before the time t is,

PN (t) =

⟨∫ t

0

dt′∫ ∞t−t′

dt′′δ

(N∑i=1

τi − t′)δ (τN+1 − t′′)

⟩

=

∫ t

0

pN (t′)dt′∫ ∞t−t′

ψ(t′′)dt′′,

P0(t) =

⟨∫ ∞t

δ(τ1 − t′)dt′⟩

=

∫ ∞t

ψ(t′)dt′, (A2)

where pN (t) =⟨δ(∑N

i=1 τi − t)⟩

is the PDF of∑Ni=1 τi.

Laplace transform of pN (t) obeys,

pN (u) =

∫ ∞0

exp[−ut]pN (t)dt =

⟨exp

(−u

N∑i=1

τi

)⟩= ψN (u), (A3)

where ψ(u) is the Laplace transform of ψ(τ). TheLaplace transform of convolution in Eq. (A2) gives,

PN (u) =ψN (u)[1− ψ(u)]

u, (A4)

where we used∫∞0ψ(τ)dτ = 1. This formula includes

N = 0 case. The normalization condition∑∞N=0 PN (t) =

1 implying∑∞N=0 PN (u) = 1/u is obeyed. The Laplace

transform of the average number of renewals 〈N(t)〉 =∑NPN (t) is,

∞∑N=0

NPN (u) =

∞∑N=1

ψN (u)

u=

ψ(u)

u[1− ψ(u)], (A5)

where we use that ψ(u) ≤ 1 with equality only at u = 0.We find

〈N(t)〉 =

∫ ε+i∞

ε−i∞

exp[ut]du

2πi

ψ(u)

u[1− ψ(u)]. (A6)

This average determines PN (t) completely because thatPN (t) is Poisson distribution with time-dependent aver-age,

PN (t)=λN (t)

N !exp[−λ(t)], λ(t) = 〈N(t)〉. (A7)

This can be verified observing that the Laplace transformof the generating function p(s) =

∑sNPn(t) is (|s| < 1)

p(u) =[1− ψ(u)]

u [1− sψ(u)](A8)

The PDF that is of interest for us here is that of the parti-cle’s displacement x(t) in time t. If N renewals occurredin time t then the displacement xN (t) obeys,

xN (t) =

N∑i=1

viτi + vN+1

(t−

N∑i=1

τi

), x0 = v1t.(A9)

Inserting 1 in P (x, t) = 〈1 × δ(x(t) − x)〉 in the formgiven by Eq. (A1) we find for the PDF of the particle’sposition that

P (x, t) = 〈δ(x(t)− x)〉 =

∫ ∞t

〈δ(v1t− x)δ(τ1 − t′)〉dt′

+

∞∑N=1

⟨∫ t

0

dt′∫ ∞t−t′

dt′′δ

(N∑i=1

τi − t′)δ (τN+1 − t′′)

δ

(N∑i=1

viτi + vN+1(t− t′)− x

)⟩. (A10)

The characteristic function P (k, t) = 〈exp[ik · x(t)]〉obeys

P (k, t) =

∫ ∞t

〈exp[ik · v1t]δ(τ1 − t′)〉dt′

+

∞∑N=1

⟨∫ t

0

dt′∫ ∞t−t′

dt′′δ

(N∑i=1

τi − t′)ψ (t′′)

exp

[N∑i=1

ik · viτi + ik · vN+1(t− t′)

]⟩. (A11)

The Laplace transform over t gives the Montroll-Weissequation in d dimensions,

P (k, u) =

⟨1− ψ(u− ik · v)

u− ik · v

⟩(A12)

+

∞∑N=1

⟨1− ψ(u− ik · v)

u− ik · v

⟩〈ψ(u− ik · v)〉N(A13)

=

⟨1− ψ(u− ik · v)

u− ik · v

⟩1

1− 〈ψ(u− ik · v)〉.(A14)

The derivations above hold irrespective of the form of thePDFs of flight times and velocities.

18

Appendix B: Small-argument expansion forlogarithm of radially symmetric Levy distribution

Here we consider the small argument behavior of Ld(x)defined by Eq. (30). Since Ld(x) is positive function withmaximum at x = 0 it can be advantageous having Taylorexpansion for lnLd(x) rather than Ld(x) itself [which isprovided by Eq. (62)]. We perform asymptotic study ofLd(r) in Eq. (30). The neighbourhood of the maximumof Ld(r) that holds at r = 0 can be described writing,

lnLd(r) = ln

[〈exp[ik · r]〉k

∫exp[−kα]

dk

(2π)d

]. (B1)

where we defined

〈exp[ik · r]〉k =

∫exp[ik · r − kα]dk∫

exp[−kα]dk, (B2)

that can be considered as average over the statistics of kdefined by the probability density function P (k),

〈exp[ik · r]〉k =

∫exp[ik · r]P (k)dk, (B3)

P (k) =exp[−kα]∫exp[−kα]dk

=αΓ(d/2)

2πd/2Γ(d/α)exp[−kα]. (B4)

The writing of the integral as average over a statisticaldistribution is useful because we can use the cumulantexpansion theorem for writing ln〈exp[ik ·r]〉k as series inr. We find

Ld(r) =21−dΓ(d/α)

πd/2αΓ(d/2)exp

[ ∞∑n=1

(−1)n〈(k · r)2n〉c,k

(2n)!

],(B5)

where c stands for cumulant (see below) and odd-ordermoments vanish because P (k) = P (−k). Quadratic cu-mulant is the dispersion,

〈(k · r)2〉c,k =rirk〈kikk〉=

r2

d〈k2〉= r2Γ[(d+ 2)/α]

dΓ(d/α).(B6)

The next non-vanishing term in the series is the quarticcumulant,

〈(k · r)4〉c,k =〈(k · r)

4〉 − 3〈(k · r)2〉2

=3r4

dΓ(d/α)

(Γ[(d+ 4)/α]

d+ 2− Γ2[(d+ 2)/α]

dΓ(d/α)

), (B7)

where we used isotropy,

〈(k · r)4〉 = rirkrprs [δikδps + δipδks + δisδpk]

〈k4〉d(d+ 2)

=3r4〈k4〉d(d+ 2)

. (B8)

The series is useful for studying the vicinity of the maxi-mum when r � 1. Higher r demand higher order cumu-lants whose form is quite cumbersome.

The cumulant expansion described above correspondsto resummation of Taylor series for Ld(r) for lnLd(r).Both series are useful when r � 1 is considered.

Appendix C: Dispersion and fourth order momentsat all times

Here we find formulas for dispersion and fourth or-der moments of the particle’s position valid at all times.These are obtained using differentiation of the Montroll-Weiss equation over k and setting k = 0. This directprocedure brings formulas that hold at arbitrary timesin contrast with the asymptotic study in the main textthat holds at large times. Thus we find more detailed in-formation on the temporal growth of the moments. Thisincludes corrections to the long-time behavior describedin the main text that can in some cases become domi-nant because of the vanishing of the leading order term.This direct calculation is getting cumbersome for higherorder moments so we perform calculation of dispersionand fourth-order moments only. Thus we describe thedifference in temporal behavior of 〈x2i (t)x2k(t)〉 for i 6= kin isotropic and XY Z... models.

The displacement’s dispersion is found from,

〈x2i 〉=−∂2

∂k2i

[⟨1−ψ(u−ik · v)

u−ik · v

⟩1

1−〈ψ(u−ik · v)〉

],

where the RHS is taken at k = 0. We find using that oddmoments of vi vanish that

〈x2i 〉 =

⟨v2i⟩

1− ψ(u)

(1− ψ(u)

u

)′′+

⟨v2i⟩ψ′′(u)

u[1− ψ(u)]

=2⟨v2i⟩

u3+

2⟨v2i⟩ψ′(u)

u2[1− ψ(u)]. (C1)

We find summing over i that,

〈x2(t)〉=2⟨v2⟩ ∫ ε+i∞

ε−i∞

exp[ut]du

2πi

uψ′(u)− ψ(u) + 1

u3[1− ψ(u)],(C2)

which can be used for finding detailed temporal depen-dence of the displacement for given ψ(τ). This formulais identical for all statistics of velocity. The leading orderterm in the limit of large times can be obtained using thesmall u-expansion ψ(u) ∼ 1− 〈τ〉u+Auα. We have

uψ′(u)− ψ(u) + 1 ∼ A(α− 1)uα, (C3)

uψ′(u)− ψ(u) + 1

u3[1− ψ(u)]= Auα−4.

Further,

ψ′(u)

ψ(u)− 1∼ −〈τ〉+Aαuα−1

−〈τ〉u+Auα=

1

u[1−Auα−1/〈τ〉]

−Aαuα−2

〈τ〉≈ 1

u− A(α− 1)uα−2

〈τ〉.

We conclude that the leading order term at small u is

〈x2i (u)〉〈v2i 〉

=〈r2(u)〉〈v2〉

∼ 2A(α− 1)uα−4

〈τ〉.

19

Thus we find in the t→∞ limit,

〈x2(t)〉 ∼2A⟨v2⟩t3−α

|Γ(1− α)|(2− α)(3− α)〈τ〉,

This reproduces the result of the main text directly fromEq. (C2). Similar conclusion is reached for

〈x4i 〉 =∂4

∂k4i

[⟨1− ψ(u− ik · v)

u− ik · v

⟩1

1− 〈ψ(u− ik · v)〉

],

where the RHS is taken at k = 0. We find using that oddmoments of vi vanish that,

〈x4i 〉 =

⟨v4i⟩

1− ψ(u)

(1− ψ(u)

u

)(4)

+6〈v2i 〉2ψ′′(u)

[1− ψ(u)]2

×(

1− ψ(u)

u

)′′+

⟨v4i⟩ψ(4)(u)

u[1− ψ(u)]+

6〈v2i 〉2[ψ′′(u)]2

u[1− ψ(u)]2.(C4)

We observe that at small u we have

1− ψ(u)

u= 〈τ〉 −Auα−1, 1− ψ(u) = 〈τ〉u−Auα,(C5)

so that

〈x4i (u)〉 ∼4A⟨v4i⟩

(−α)4uα−6

〈τ〉α, (C6)

where the neglected terms involving 〈v2i 〉2 are propor-tional to u2α−7 and can be neglected at small u becauseof α > 1. This reproduces the result of the main text,

〈x4i (t)〉 ∼4A⟨v4i⟩

|Γ(1− α)|(4− α)(5− α)〈τ〉t5−α. (C7)

We consider the cross-correlation 〈x2ix2k〉. We have

〈x2i (t)x2k(t)〉 =∂4

∂k2i ∂k2k

[⟨1− ψ(u− ik · v)

u− ik · v

⟩× 1

1− 〈ψ(u− ik · v)〉

],

where the RHS is taken at k = 0. We find using thatodd moments of vi, vk vanish that

〈x2i (t)x2k(t)〉 =

⟨v2i v

2k

⟩1− ψ(u)

(1− ψ(u)

u

)(4)

+2〈v2i 〉〈v2k〉ψ′′(u)

[1− ψ(u)]2

×(

1− ψ(u)

u

)′′+

⟨v2i v

2k

⟩ψ(4)(u)

u[1− ψ(u)]+

2〈v2i 〉〈v2k〉[ψ′′(u)]2

u[1− ψ(u)]2.

This can be used for detailed study of temporal behaviorof 〈x2i (t)x2k(t)〉. We use this formula for finding the lead-ing order long-time behavior in XY Z... model. There wehave

⟨v2i v

2k

⟩= 0 for i 6= k, 〈v2i 〉 = v20/d which gives

〈x2i (t)x2k(t)〉 =2v40ψ

′′(u)

[1− ψ(u)]2d2

[(1− ψ(u)

u

)′′+ψ′′(u)

u

].

The leading order term when u is small is

〈x2i (t)x2k(t)〉 =4v40α(α− 1)2A2u2α−7

〈τ〉2d2. (C8)

Performing inverse Laplace transform we find

〈x2i (t)x2k(t)〉 =4v40α(α− 1)2A2

Γ(7− 2α)〈τ〉2d2t6−2α. (C9)

The probability distribution is characterized by constanttime-independent ratio

〈x2i (t)x2k〉〈x2i (t)〉〈x2k(t)〉

=4αΓ2(2− α)

Γ(7− 2α), (C10)

telling that interdependence of the displacement’s com-ponents becomes constant at large times. In contrastusing

⟨v4i⟩

= v40/d (this can be found from v40 =⟨(∑v2i)2⟩

= d〈v4i 〉 where we use that cross-correlations

of velocity vanish) we find that the ratio

〈x2i (t)x2k(t)〉〈x4i (t)〉+ 〈x4k(t)〉

=α(α− 1)(4− α)(5− α)AΓ(2− α)

(2d)〈τ〉Γ(7− 2α)tα−1,

decreases with time indefinitely characterizing growinganisotropy of the distribution.

Appendix D: Infinite density is the generatingfunction of the moments

We demonstrate that moments of integer order are de-scribed by the infinite density. We observe that Eqs. (87)-(88) give for the Fourier transform of I(v) that,

I(k) =A

|Γ(1− α)|〈τ〉

∞∑n=1

(2n)(−1)n⟨(k·v)2n

⟩(2n)!(2n− α)(2n+ 1− α)

.

Comparing this with Eq. (103) we conclude that in thelimit of large times,

〈xi1(t)xi2(t) . . . xi2n(t)〉t2n+1−α ∼

∫vi1vi2 . . . vi2nI (v) dv.

This implies that,

〈xi1(t)xi2(t) . . . xi2n(t)〉 ∼∫xi1xi2 . . . xi2n

td+α−1I(xt

)dx,

that is P (x, t) provided by Eq. (97) describes the long-time limit of the moments. In one-dimensional case thisresult was derived in [17].

Appendix E: Fractional derivative form of Fouriertransform and isotropic Levy distributions

In the recent work [21] numerical study of two-dimensional Levy distributions was performed observ-ing that the distribution can be written as fractional

20

derivative of the distribution in one dimension and us-ing the Matlab code for the fractional derivative. Herewe observe that this consideration has universal appli-cablility. We demonstrate that d−dimensional Fouriertransform of arbitrary radially symmetric function canbe written as fractional derivative of order (d − 1)/2of the one-dimensional Fourier transform of that func-tion. We clarify that this fact deserves to be knownbecause it gives simple way of studying d−dimensionaltransforms based on simpler one-dimensional transform.Series expansions are obtained immediately using frac-tional derivatives of powers if term-by-term differentia-tion of series for one-dimensional Fourier transform isvalid. Similarly the asymptotic form of the transform atlarge argument can be found by differentiation of sim-pler one-dimensional form. Numerically d−dimensionaltransforms are obtained applying fractional derivativecode on well-developed one-dimensional Fourier trans-form code.

We do the calculations for inverse Fourier transformwith formulas for direct transform implied. We considerthe d−dimensional Fourier transform of radially symmet-ric function fd(x),

f(k) =

∫exp[−ik · x]fd(x)dx, (E1)

where |x| = x. The formula for the inverse Fourier trans-form of radially symmetric function whose Fourier trans-form is f(k) where k = |k| is

fd(x) =x1−d/2

(2π)d/2

∫ ∞0

Jd/2−1(kx)kd/2f(k)dk. (E2)

We introduce fd(x) = fd(x2) where

fd(x) =x1/2−d/4

(2π)d/2

∫ ∞0

Jd/2−1(k√x)kd/2f(k)dk, (E3)

where d is the dimension of space. We find the depen-dence of fd on d when f(k) is fixed function. We observethat operator of fractional derivative of order 1/2 whoseaction on arbitrary well-behaved function h obeys,

D1/2− h = − d

dx

1√π

∫ ∞x

h(x′)dx′√x′ − x

, (E4)

is dimension raising,

1√πD

1/2− fd = fd+1. (E5)

We use that the identity,

d

dx

[x−νJν(x)

]= −x−νJν+1(x), (E6)

implies

d

dx

[x−ν/2Jν(k

√x)]

= −kx−(ν+1)/2Jν+1(k

√x)

2.(E7)

We observe that D1/2− can be written in terms of the

right-side fractional integral I1/2− defined as [43],

I1/2− [h] =

1√π

∫ ∞x

h(x′)dx′√x′ − x

. (E8)

We have

D1/2− [h] = −I1/2− [h′], (E9)

where we use that,

d

dx

∫ ∞x

h(x′)dx′√x′ − x

= 2d

dx

∫ ∞x

h(x′)dx′d

dx′√x′ − x

= −2d

dx

∫ ∞x

h′(x′)dx′√x′ − x =

∫ ∞x

h′(x′)dx′√x′ − x

.(E10)

We find using fractional integral from p. 205 in [42] that,

1√πD

1/2−

[x−ν/2Jν(k

√x)]

=k

2π

∫ ∞x

x′−(ν+1)/2dx′√x′ − x

×Jν+1(k√x′) =

√k

2πx−(ν+1/2)/2Jν+1/2(k

√x). (E11)

Thus we find Eq. (E5) by acting on Eq. (E3) with D1/2−

and using identity (E11). Further applying d − 1 times

D1/2− on f1 and using Eq. (E5) we obtain that,

1

π(d−1)/2

[D

1/2−

]d−1f1 = fd. (E12)

If D1/2− would be ordinary derivative then the above

would imply

1

π(d−1)/2D(d−1)/2− f1 = fd, (E13)

where the fractional derivative of order α is,

Dα−f =

(−1)n

Γ(n− α)

dn

dxn

∫ ∞x

f(x′)dx′

(x′ − x)α−n+1, (E14)