planar maps, random walks and the circle packing · pdf fileplanar maps, random walks and the...

Planar maps, random walks and the circle packingtheorem

Asaf NachmiasTel-Aviv University

Informal analysis seminar, Kent State, April 30th 2016

Asaf Nachmias Planar maps, random walks and the circle packing theorem

Basic terminology

Planar map: a graph embedded in R2 so that vertices aremapped to points and edges to non-intersecting curves.

(a) Triangulation: each face has three edges.(b) Quadrangulation: each face has four edges.

The simple random walk on a graph starts at an arbitraryvertex and in each step moves to a uniformly chosen neighbor.

Basic terminology

The basic questions

Is the random walk recurrent or transient?

Are there bounded harmonic functions on the graph?

What is the typical distance of the walker from the startingpoint?

These are geometric/probabilistic questions, intimately related tovolume growth, isoperimetric profile, and resistance growth.

Lastly, can we analyze models of statistical physics on suchgraphs? (percolation, self-avoiding walk, Ising model)

The basic questions

Random planar maps

Let Gn be a uniform random triangulation or quadrangulation on nvertices. [Efficient sampling: Tutte’s enumeration or Schaeffer’sbijections.]

(image by Nicolas Curien)

When n→∞, what does it look like?

Random planar maps

(Meta-) Physics

In physics, the study of such random triangulation is calledquantum gravity. The essential idea is to extend to higherdimensions the concept of Feynman integrals on paths.Triangulations are used as discretized version of a 2 dimensionalmanifold, and a function is summer over all of them. Physicists areeven more interested in a continuous scaling limit of the discretizedmodel.

The KPZ formula (Knizhnik, Polyakov, Zamolodchnikov ’88)relates exponents for random processes (random walk, criticalpercolation, self-avoiding walk, critical Ising model etc.) in theEuclidean lattice to the corresponding exponents in quantumgravity (which are often much easier to calculate).

(Meta-) Physics

In physics, the study of such random triangulation is calledquantum gravity. The essential idea is to extend to higherdimensions the concept of Feynman integrals on paths.Triangulations are used as discretized version of a 2 dimensionalmanifold, and a function is summer over all of them. Physicists areeven more interested in a continuous scaling limit of the discretizedmodel.

The KPZ formula (Knizhnik, Polyakov, Zamolodchnikov ’88)relates exponents for random processes (random walk, criticalpercolation, self-avoiding walk, critical Ising model etc.) in theEuclidean lattice to the corresponding exponents in quantumgravity (which are often much easier to calculate).

The KPZ formula (pure gravity form)

If a set of a random process on a fixed 2D regular lattice hasdimension 2(1− x) and the corresponding set on the randomlattice has dimension 4(1−∆), then the KPZ relation states that

x =∆(2∆ + 1)

Example: consider the typical displacement of the self-avoidingwalk of length t (that is, choose a uniform non-intersecting path oflength t).

On a random map it is not so difficult to show that thisdisplacement is t1/2, corresponding to ∆ = 1/2.

By KPZ, x = 1/3 corresponding to displacement of t3/4 onthe Euclidean lattice. Proving the latter is a notoriouslyhard problem!

x =∆(2∆ + 1)

By KPZ, x = 1/3 corresponding to displacement of t3/4 onthe Euclidean lattice.

Proving the latter is a notoriouslyhard problem!

x =∆(2∆ + 1)

How to make this science fiction into reality?

Duplantier and Sheffield (2011) provided a rigorous KPZcorrespondence between:

1 Lebesgue measure L on the 2-sphere S2 and

2 a random measure µ on S2 which can be formally written asdµdL = eγh where γ > 0 is a constant and h is an instance of aGaussian Free Field.

The missing piece of the puzzle: does the area measure of somenatural embedding in S2 of a random map Gn converges to µ?

The bottom line

We need to understand random planar maps from different pointof views:

Local vs. global structure,

Random metric space vs. random Riemannian surface.

Statistical physics models on random planar graphs(self-avoiding walk, percolation, Ising...)

There are several appealing ways to embed a map:

1 The conformal embedding,

2 The harmonic embedding,

3 Via Koebe’s circle packing theorem.

The bottom line

−8e+06 −6e+06 −4e+06 −2e+06 0e+00 2e+06 4e+06 6e+06

−6e+

06−4

−2e+

Re(bbox(self$pos, max(self$r)))

Image by Maxim Krikun

Circle packing

Let G be a finite simple planar graph.

What would be a nice (canonical) way of drawing G in the plane?

Theorem (Koebe 1936, Andreev 1970, Thurston 1985)

A finite simple planar graph is a tangency graph of a circle packing.

If G is a triangulation, then the drawing is unique up to Mobiustransformations and reflections.

Circle packing

Infinite circle packings

How does the theory extend to infinite graphs?

A circle packing always exists (as long as G has finite degrees)as one can exhaust the graph and take some limit of the finitecircle packings.

No rigidity: different limits may lead to very different lookingpackings (that are not Mobius equivalent). However, undersome natural conditions, the type of the limiting packing isdetermined.

The main point: the type of the packing encapsulatesprobabilistic information: recurrence/transience of the randomwalk, existence of non-trivial bounded harmonic functions,speed of the random walk, etc.

No rigidity: different limits may lead to very different lookingpackings (that are not Mobius equivalent).

However, undersome natural conditions, the type of the limiting packing isdetermined.

Circle packing definitions

A circle packing P = {Cv} is a set of circles in the planewith disjoint interiors.

The tangency graph of P is a graph G (P) in which thevertex set is the set of circles, and two circles are adjacentwhen they are tangent.

An accumulation point of P is a point y ∈ R2 such thatevery neighborhood of it intersects infinitely many circles of P.

From now on we assume that G is an infinite triangulation.The carrier of P is the union over all faces (except for theouter face, if it exists) of the three circles of the face togetherwith the bounded space between them (the interstice).

The set of accumulation points A(P) is the boundary of thecarrier carr(P).

From now on we assume that G is an infinite triangulation.

The carrier of P is the union over all faces (except for theouter face, if it exists) of the three circles of the face togetherwith the bounded space between them (the interstice).

Example 1

The triangular lattice.

Here carr(P) = R2 and A(P) = ∅.

Example 2

The 7-degree hyperbolic tessellation circle packed in the unit disc.

Here carr(P) = {z ∈ C : |z | < 1} and A(P) = {z ∈ C : |z | = 1}.

Example 3

The 7-degree hyperbolic tessellation circle packed in the unitsquare.

−8e+06 −6e+06 −4e+06 −2e+06 0e+00 2e+06 4e+06 6e+06

−6e+

06−4

−2e+

Re(bbox(self$pos, max(self$r)))

Image by Maxim Krikun

The He-Schramm Theorem (1995)

A graph G is one-ended if the removal of any finite set leavesprecisely one infinite component.

If P is a circle packing of atriangulation, then G (P) is one-ended iff carr(P) is simplyconnected.

Theorem (He-Schramm ’95)

Assume that G is a bounded degree, one-ended triangulation andlet P be a circle packing of it.

1 If carr(P) = R2, then G is recurrent.

2 If carr(P) 6= R2, then G is transient. Moreover,

3 If G is transient, then for any simply connected domainD 6= R2 there exists a packing P such that G (P) = G andcarr(P) = D.

A graph G is one-ended if the removal of any finite set leavesprecisely one infinite component. If P is a circle packing of atriangulation, then G (P) is one-ended iff carr(P) is simplyconnected.

2 If carr(P) 6= R2, then G is transient.

Moreover,

Bounded degree in necessary

More examples

The 7-degree hyperbolic half-space glued with the 6-degreetriangular half-space.

More examples

Rings of degree 7 (grey) are separated by growing bands of degreesix vertices (white), causing the hyperbolic radii of circles to decay.The bands of degree six vertices can grow surprisingly quicklywithout the triangulation becoming recurrent.

The scaling limit of random planar maps

At the large scale a random planar map looks like this:

(image by Jean-Francois Marckert)

This is what you get when you scale graph distances by a factor ofn−1/4 so that the diameter is roughly a constant, then take aGromov-Hausdorf limit. The resulting limit is called the Brownianmap and is a random compact metric space homeomorphic to thesphere (Le-Gall 2011, Miermont 2011).

At the large scale a random planar map looks like this:

Local limits

The other extremal option is: don’t scale distances and aim to getan infinite graph in the limit.

Definition. Given a sequence of finite graphs Gn let ρn be auniform random vertex of Gn. We say that the random rootedgraph (G , ρ) is the distributional limit of Gn if for any r > 0

BGn(ρn, r)(d)

=⇒ BG (ρ, r) ,

where BG (ρ, r) is the ball of radius r in graph distance around ρ.

Local limits

The other extremal option is: don’t scale distances and aim to getan infinite graph in the limit.

Definition. Given a sequence of finite graphs Gn let ρn be auniform random vertex of Gn. We say that the random rootedgraph (G , ρ) is the distributional limit of Gn if for any r > 0

BGn(ρn, r)(d)

=⇒ BG (ρ, r) ,

where BG (ρ, r) is the ball of radius r in graph distance around ρ.

Examples

Gn = path of length n =⇒

Gn = G (n, cn ) =⇒ Galton-Watson tree with offspringdistribution Poisson(c).

Gn = binary tree of height n =⇒ .

Examples

Gn = path of length n =⇒ Z.

Examples

Gn = G (n, cn ) =⇒

Galton-Watson tree with offspringdistribution Poisson(c).

Examples

Gn = binary tree of height n =⇒

Examples

Gn = binary tree of height n =⇒ infinite binary tree.

Examples

Gn = binary tree of height n =⇒ infinite binary tree.

Examples

Gn = binary tree of height n =⇒ the canopy tree.

Examples

Gn = binary tree of height n =⇒ the canopy tree.

The uniform infinite planar triangulation/quadrangulation

Let Gn be a uniform random triangulation or quadrangulation on nvertices. Does it have a distributional limit?

Theorem (Angel-Schramm 2003, Krikun 2005)

The limit exists, and it is an infinite triangulation/quadrangulationof the plane.

The limit is commonly known as the uniform infinite planartriangulation/quadrangulation (UIPT/UIPQ).

Universality: Other models of random planar graphs are expectedto have a distributional limit with similar properties.

Properties of the UIPT/UIPQ vs. Euclidean geometry

The UIPT/UIPQ behaves like Z2 in some aspects:

One-ended (Angel-Schramm 2003).

pc = 12 (Angel 2003).

Liouville, that is, no bounded nonconstant harmonic functions(Benjamini-Curien 2010).

but not all aspects:

Unbounded degrees.

Balls of radius r have size roughly r4 (Angel 2003,Chassaing-Schaeffer 2004).

Strictly sub-diffusive (Benjamini-Curien 2012).

Question: is it recurrent or transient a.s.?

pc = 12 (Angel 2003).

Unbounded degrees.

pc = 12 (Angel 2003).

Unbounded degrees.

pc = 12 (Angel 2003).

Unbounded degrees.

pc = 12 (Angel 2003).

Unbounded degrees.

pc = 12 (Angel 2003).

Unbounded degrees.

pc = 12 (Angel 2003).

Unbounded degrees.

pc = 12 (Angel 2003).

Unbounded degrees.

pc = 12 (Angel 2003).

Unbounded degrees.

Recurrence?

Conjecture (Benjamini-Schramm 2001, Angel-Schramm 2003)

The UIPT/UIPQ is almost surely recurrent.

If only the UIPT had bounded degrees...

Theorem (Benjamini-Schramm 2001)

Any distributional limit of finite, planar graphs with boundeddegrees is almost surely recurrent.

Examples: Z, Z2, the canopy tree.Remark: Bounded degree is a necessary condition:

Examples: Z, Z2, the canopy tree.

Remark: Bounded degree is a necessary condition:

1 1 1 1 1 1 1 1

Recurrence criterion for planar graph limits

Theorem (Gurel-Gurevich & N. 2013)

Any distributional limit of finite, planar graphs where the degree ofthe root has an exponential tail is almost surely recurrent.

Corollary (Gurel-Gurevich & N. 2013)

Indeed, it is known that the degree of the root of the UIPT/UIPQhas an exponential tail (Angel-Schramm 2003, Benjamini-Curien2012).

Sharpness: slightly fatter tail than exponential is not enough.

The plan:

1 Crash course on electric networks and their probabilisticinterpretation.

2 The He-Schramm Theorem (part 1 and 2 only).

3 Recurrence of random planar maps.

4 Time permitting:

The He-Schramm theorem in multiply connected domain andconnections to quasi-conformal maps (joint withGurel-Gurevich and Souto).

Boundary theory for transient planar maps (joint with Angel,Barlow, Gurel-Gurevich).

Uniform spanning forests of planar maps are connected (jointwith Hutchcroft).

The plan:

4 Time permitting:

The plan:

4 Time permitting:

The plan:

4 Time permitting:

The plan:

4 Time permitting:

The plan:

4 Time permitting:

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