# 44th Annual John H. Barrett Memorial Lectures

## Complex Analysis in Probabilistic Settings

### June 16-19, 2014

#### Survey Series Lectures by:

##### Grégory Miermont, ENS de Lyon
• Random planar maps I: trees and maps
• Random planar maps II: local limits
• Random planar maps III: scaling limits
##### Steffen Rohde, University of Washington
• Random curves and conformal welding
• Conformal representation of planar maps
• Random surfaces and the conformal loop ensemble

Additional Speakers Include: Omer Angel, Ilia Binder, Mario Bonk, Edward Crane, James Gill, Greg Lawler, Jason Miller, Pietro Poggi-Corradini, Ken Stephenson, Amanda Turner, Brent Werness

## Abstracts

Grégory Miermont
Talk 1: Random planar maps I: Trees and maps
Talk 2: Random planar maps II: Local limits
Talk 3: Random planar maps III: Scaling limits
Abstract: A (planar) map is a graph embedded in the 2-sphere, considered up to homeomorphisms. In a way, it provides a discrete geometrization on the sphere. Thus, a map chosen at random in some sense is a natural candidate for a notion of a spherical geometry chosen at random. More precisely, one expects that, as
the mesh of the map refines while the number of vertices blows up, random maps approximate a random surface, in the same way as random walks are the natural discrete approximations of Brownian motion.

Like Brownian motion, the continuum surfaces that arise in this context are very wild, and far from being smooth Riemannian structures. It makes their study quite interesting, in the sense that one has to look for the geometric notions that still make sense in this context, such as distances and geodesics. By contrast with these “continuum” notions, we will see that the study of maps relies on tools that are of purely combinatorial nature, with an emphasis on the so-called Schaeffer bijection allowing to see maps as decorated trees. We will also discuss the conjectural connection of random maps with random conformally invariant objects.

I. Trees and maps: We will see how labeled trees are natural coding objects for maps via the celebrated Schaeffer bijection and its variants, and will give the first limiting results in the context of labeled trees.

II. Local limits: Following the pioneering approach by Angel and Schramm, we will consider the infinite-volume limit of random maps in arbitrary but fixed neighborhoods of a given vertex. After giving several possible constructions of these spaces, we will briefly discuss some of their important properties, in particular some aspects of their geometry at infinity through the Gromov boundary, or the fact that they form recurrent graphs (a recent theorem by Gurel-Gurevich and Nachmias).

III. Scaling limits: Following a different direction initiated by Chassaing-Schaeffer, we will show how to renormalize distances in maps so that they converge to a continuum scaling limit, called the Brownian map. We will discuss the topological and metric properties of this object, and state a couple of conjectures pertaining to a ”random conformal” description of this object.

Steffen Rohde
Talk 1: Random curves and conformal welding
Talk 2: Conformal representation of planar maps
Talk 3: Random surfaces and the conformal loop ensemble
Abstract: In the first talk, I will briefly review several ways to generate random conformal maps, with an emphasis on the Schramm-Loewner evolution SLE and its natural appearance as the scaling limit of various two-dimensional lattice models. I will then focus on conformal welding, going from basic properties to its connections to SLE.

In the second talk, I will discuss circle packings and Belyi functions as conformally natural embeddings of planar maps into the euclidean plane. An emphasis will be on trees (maps with only one face) and their representation through Shabat polynomials.

The third talk is devoted to questions regarding conformal uniformization of surfaces and of carpets. In particular we will discuss an analog of Bonk’s Sierpinski carpet uniformization in the setting of the Conformal Loop Ensemble CLE, based on joint work with Brent Werness.

Omer Angel
Title: Random walks on transient planar graphs
Abstract: I will describe results on random walks on transient planar graphs. In the general case, I will describe a connection between the Poisson and Martin boundaries and the topological boundary obtained by circle packing the graph in the unit disk. (Joint with Martin Barlow, Ori Gurel-Gurevich and Asaf Nachmias).

Ilia Binder
Title: Computability and Conformal Mappings
Abstract: I will discuss recent results on computability of conformal mappings and their boundary extensions. In particular, I will discuss the conditions for computability of the harmonic measure. I will also talk about computability questions arising in Complex Dynamics.

Mario Bonk
Title: Parabolicity of leaves
Abstract: Certain dynamical systems give rise to foliations where the leaves are quasi-isometric to open simply connected surfaces. The question arises whether these leaves are parabolic or hyperbolic (equivalent to recurrence or transience of a random walk). This is related, for example, to Cannon’s conjecture in geometric group theory or to Thurston’s characterization of postcritically-finite rational maps. In my talk I will discuss some background and open problems in this area.

Edward Crane
Title: Renewals, forest fires and continuum random trees
Abstract: Analysis has long been a central tool for understanding probabilistic models, both in discrete and continuous time, so there are now many different connections between analysis and probability. In this lecture we will look at two examples of probabilistic models that can be understood very precisely using Laplace transforms.

The first example is the renewal process with U[0,1] inter-arrival times, which we motivate by looking at an urn model called the simple harmonic urn, studied in joint work with Nic Georgiou, Stas Volkov, Andrew Wade and Rob Waters. Careful analysis of the the Laplace transform gave us sharp asymptotics for the first two moments of the number of renewals up to time t, much sharper than we actually needed!

The second example is a so-called forest fire model on the complete graph on n vertices. This model was shown to exhibit self-organized criticality in the large n limit by Balazs Rath and Balint Toth, by a careful analysis of the controlled inviscid Burgers equation that is satisfied by the Laplace transform of the size-biased cluster size distribution. In recent joint work with Nic Freeman and Balint Toth we have proved a local limit theorem for this model. Our analysis gives the probabilistic interpretation of survival time distributions to the characteristic curves
that were used to understand the PDE. We also show that in a stationary version of the same model the large clusters rescale to the Brownian continuum random tree; the proof is analytic, using the Cramer-Wold device and the moment method.

James Gill
Title: Local Limits, Doubling Metric Spaces, and a Lemma
Abstract: In 2001 I. Benjamini and O. Schramm proved that the distributional limit of a random graph with bounded degree is almost surely recurrent with respect to a random walk. To prove this fact they devised an interesting lemma about finite point sets in the plane. Since then this same lemma has been used several times in different contexts. In particular, in joint work with S. Rohde, we used it to show that the uniform infinite planar triangulation is almost surely a parabolic Riemann surface. In further investigation by the speaker it has recently been found that the conclusion of this lemma holds in any doubling metric space. In fact, the metric doubling condition is equivalent to the property described in the lemma.

Gregory Lawler
Title: Conformal invariance of the Green’s function for loop-erased random walk
Abstract: If D is a simply connected complex domain containing the origin with two boundary points a, b and $$A_n$$ is an approximating lattice domain, we consider the probability that the loop-erased random walk (LERW) in $$A_n$$ from (points near to) a to b goes though the origin. We show that the normalized limit is given by the Schramm-Loewner evolution (SLE) Green’s function prediction — indeed, there exists $$c_0$$, u such that the probability is $$c_0(rn)^{-3/4} [sin^3 \theta + O(n^{-u})]$$ where r is the conformal radius of D with respect to the origin and $$\theta$$ is the argument of the origin in D with respect to a,b. I will discuss the relationship between this and questions about natural parametrization of LERW and SLE. This is joint work with Christian Benes and Fredrik Viklund.

Jason Miller
Title: Random Surfaces and Quantum Loewner Evolution
Abstract: We will describe a new universal family of growth processes called Quantum Loewner Evolution (QLE) and explain how QLE can be used to relate $$\sqrt{8/3}$$-Liouville quantum gravity with the Brownian map. We will also explain how QLE is related to diffusion limited aggregation, first passage percolation, and the dielectric breakdown model.

Title: Modulus of families of walks and applications
Abstract: How far apart are two subsets of nodes in a graph? How rich is the family of walks that go from point a to point b while visiting point c? How much damage can be created by an epidemics that originates in x before it reaches y? Such questions and more can be tackled by computing the modulus of an appropriate family of walks. In this talk we will show how modulus generalizes and extends more common notions such as shortest paths, minimal cuts, or effective resistances (when thinking of a graph as an electrical network). Modulus of families of walks is a concept that originated in complex analysis and geometric function theory and is therefore a very well-developed tool with a rich history of theoretical uses. In recent years, modulus has been considered on graphs as well, in the work of Oded Schramm for instance. We will show how it can used to provide both theoretical and real-world applications.

Ken Stephenson
Title: Conformal Tilings: Theory and Examples
Abstract: (Joint work with Phil Bowers, Florida State) This talk will introduce the theory of “conformal” tiling and illustrate several examples, concentrating in particular on conformal versions of traditional heierarchical aperiodic tilings such as the Penrose, pinwheel, chair, domino, and sphinx tilings. Emphasis will be on experimentation and observation. (Software will be running in the Demo Session.) Questions of type, rigidity, function theory, and shape will be discussed.

Amanda Turner
Title: Small-particle limits in a regularized Laplacian random growth model
Abstract: In 1998 Hastings and Levitov proposed a one-parameter family of models for planar random growth in which clusters are represented as compositions of conformal mappings. This family includes physically occurring processes such as diffusion-limited aggregation (DLA), dielectric breakdown and the Eden model for biological cell growth. In the simplest case of the model (corresponding to the parameter $$\alpha$$= 0), James Norris and I showed how the Brownian web arises in the limit resulting from small particle size and rapid aggregation. In particular this implies that beyond a certain time, all newly aggregating particles share a single common ancestor. I shall show how small changes in alpha result in the emergence of branching structures within the model so that, beyond a certain time, the number of common ancestors is a random number whose distribution can be obtained. This is based on joint work with Fredrik Johansson Viklund (Uppsala) and Alan Sola (Cambridge).

Brent Werness
Title: Alternative constructions of the Gaussian free field and fast simulation of Schramm-Loewner evolutions
Abstract: The Schramm–Loewner evolutions (SLE) are a family of stochastic processes which describe the scaling limits of curves which occur in two-dimensional critical statistical physics models. SLEs have had found great success in this task, greatly enhancing our understanding of the geometry of these curves. Despite this, it is rather dicult to produce large, high-fidelity simulations of the process due to the significant correlation between segments of the simulated curve. The standard simulation method works by discretizing the construction of SLE through the Loewner ODE which provides a quadratic time algorithm in the length of the curve.

Recent work of Sheeld and Miller has provided an alternate description of SLE, where the curve generated is taken to be a flow line of the vector field obtained by exponentiating a Gaussian free field. In this talk, I will describe a new method of approximately sampling a Gaussian free field, and show how this allows us to more eciently simulate an SLE curve.

## Participants

Conference will be held on the University of Tennessee campus located in Knoxville, Tennessee.

Organizing Committee: Drs. Joan Lind (Chair) and Ken Stephenson

This conference is being sponsored by NSF, The University of Tennessee Mathematics Department,
College of Arts & Sciences, and the Office of Research.