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Timetable (RGMW05)

Geometry of random walks and SLE: a birthday conference for Greg Lawler

Monday 15th June 2015 to Friday 19th June 2015

Monday 15th June 2015
09:00 to 09:50 Registration
09:50 to 09:55 Welcome from Christie Marr (INI Deputy Director)
09:55 to 10:00 Welcome from Organisers
10:00 to 11:00 W Werner (ETH Zürich)
Some news from the loop-soup front
Co-author: Wei Qian (ETH Zürich)

We will present some new features of loop-soups.

INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Conformal representations of Random Maps and Surfaces
Motivated by the quest to find a somewhat conformal map from the Riemann sphere to a Liouville quantum-gravity sphere, I will talk about uniformization of discrete maps (based on joint work with Don Marshall), and discuss an analog of Mario Bonk's carpet uniformization in the setting of the Conformal Loop Ensemble CLE (based on joint work with Brent Werness).
INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 J Miller (Massachusetts Institute of Technology)
Liouville quantum gravity and the Brownian map
Co-author: Scott Sheffield (MIT)

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has roots in string theory and conformal field theory. The second is the Brownian map, which has roots in planar map combinatorics. We show that the Brownian map is equivalent to Liouville quantum gravity with parameter $\gamma = \sqrt{8/3}$.

INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 Essential spanning forests on periodic planar graphs
The laplacian on a periodic planar graph has a rich algebraic and integrable structure, which we usually don't see when we do standard potential theory. We discuss the combinatorial, algebraic and integrable features of the laplacian, and in particular interpret combinatorially the points of the "spectral curve" of the laplacian in terms of probability measures on spanning trees and forests.
INI 1
16:30 to 17:30 Welcome Wine Reception
Tuesday 16th June 2015
09:00 to 10:00 Competitive erosion is conformally invariant
Co-author: Shirshendu Ganguly (University of Washington)

We study a graph-theoretic model of interface dynamics called {\bf competitive erosion}. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective sources and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. This is a finite competitive version of the celebrated Internal DLA growth model first analyzed by Lawler, Bramson and Griffeath in 1992. We establish conformal invariance of competitive erosion on discretizations of smooth, simply connected planar domains. This is done by showing that at stationarity, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and more generally by a hyperbolic geodesic. (Joint work with Shirshendu Ganguly, available at http://arxiv.org/abs/1503.06989 ).

INI 1
10:00 to 10:10 Break
10:10 to 11:10 A-S Sznitman (ETH Zürich)
On Disconnection, random walks, random interlacements, and the Gaussian free field
In this talk we will discuss some large deviations estimates related to the question of understanding how a simple random walk in dimension 3 and above can insulate a macroscopic body. Some of these results have been obtained in collaboration with Xinyi Li.
INI 1
11:10 to 11:30 Morning Coffee
11:30 to 12:30 tba INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 B Werness (University of Washington)
Convergence of discrete holomorphic functions on non-uniform lattices
The theory of discrete holomorphic functions has been studied by researchers from a diverse set of fields from classical complex analysts to applied computer scientists. In the field of conformally invariant random processes, discrete analyticity has found a particularly central role as the convergence of discrete analytic functions to their continuum counterparts is the key step in the showing convergence of discrete random processes to Schramm--Loewner Evolutions.

In this talk, we will discuss recent work that proves that discrete analytic functions converge to their continuum counterparts on lattices with only local control on the geometry. We will then discuss potential applications of this result to the study conformally invariant random processes on random surface models.

INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 Conformal restriction: the chordal and the radial
Co-authors: Greg Lawler (Math. Department of Chicago University), Oded Schramm (Microsoft Research), Wendelin Werner (Math. Department of ETH)

When people tried to understand two-dimensional statistical physics models, it is realized that any conformally invariant process satisfying a certain restriction property has corssing or intersection exponents. Conformal field theory has been extremely successful in predicting the exact values of critical exponents describing the bahvoir of two-dimensional systems from statistical physics. The main goal of this talk is to review the restriction property and related critical exponents. First, we will introduce Brownian intersection exponents. Second, we discuss Conformal Restriction---the chordal case and the radial case. Third, we explain the idea of the proofs. Finally, we give some relation between conformal restriction sets and intersection exponents.

INI 1
Wednesday 17th June 2015
09:00 to 10:00 Scaling window of Bernoulli percolation on Z^d
We will discuss the notion of scaling window for Bernoulli percolation on Z^d. While the notion is classical in two dimensions, the systematic study of the size of the window in higher dimension has not been done by now. We will present a few progress in this direction.
INI 1
10:00 to 10:10 Break
10:10 to 11:10 A (slightly) new look at the backbone
Co-author: Jean-Christophe MOURRAT (ENS Lyon)

In this talk, I will start by discussing one of Greg's many influential papers: "One-arm exponent for critical 2D percolation" by Lawler, Schramm and Werner. I will then focus on the particular case of the backbone exponent with some new insights on this topic. Joint work with Jean-Christophe Mourrat.

INI 1
11:10 to 11:30 Morning Coffee
11:30 to 12:30 A random walk proof of Kirchhoff's matrix tree theorem
Kirchhoff's matrix tree theorem relates the number of spanning trees in a graph to the determinant of a matrix derived from the graph. There are a number of proofs of Kirchhoff's theorem known, most of which are combinatorial in nature. In this talk we will present a relatively elementary random walk-based proof of Kirchhoff's theorem due to Greg Lawler which follows from his proof of Wilson's algorithm. Moreover, these same ideas can be applied to other computations related to general Markov chains and processes on a finite state space. Based in part on joint work with Larissa Richards (Toronto) and Dan Stroock (MIT).
INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 Return of the Multiplicative Coalescent
Co-authors: David Aldous (UC Berkeley), Mathieu Merle (Université Paris 7), Justin Salez (Université Paris 7)

No one steps in the same river twice. Sometimes life makes a (almost) second chance possible. I will talk about my thesis work. The choice of the topic will be explained at the talk.

INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 Where Planar Simple Random Walk Loses its Rotational Symmetry
Local central limit theorems for random walks come in several shapes and forms. Lawler's books contain a number of versions which are very precise at and somewhat beyond the typical range for random walk. In this talk, I will give a detailed description of what happens for a large range of atypical points for simple random walk and will show where and how the walk loses its approximate rotational symmetry.
INI 1
19:30 to 22:00 Conference Dinner at Gonville and Caius College
Thursday 18th June 2015
10:00 to 11:00 SLE Quantum Multifractality INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Radial SLE martingale-observables
Co-author: Nikolai Makarov (Caltech)

After implementing a version of radial conformal field theory in the OPE family of statistical fields generated by background charge modification of the Gaussian free field, I present an analytical and probabilistic proof of a well-known statement in physics that the correlation functions of such fields under the insertion of one-leg operator form a collection of radial SLE martingale-observables. In the construction of one-leg operator, the so-called neutrality conditions on the charges play an important role. I explain two neutrality conditions: first, the linear combination of bosonic fields is to be a well-defined Fock space field; second, the Coulomb gas correlation function is to be conformally invariant. To reconcile these two neutrality conditions, one needs to place the background charge at the marked interior point, the target point of SLE. This is a joint work with Nikolai Makarov.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 Scaling limit of the probability that loop-erased random walk uses a given edge
Co-authors: Christian Benes (CUNY), Greg Lawler (University of Chicago)

I will discuss a proof of the following result: The probability that a loop-erased random walk (LERW) uses a given edge in the interior of a lattice approximation of a simply connected domain converges in the scaling limit to a constant times the SLE(2) Green's function, an explicit conformally covariant quantity. I will also indicate how this result is related to convergence of LERW to SLE(2) the natural parameterization.

This is based on joint work with Christian Benes and Greg Lawler and work in progress with Greg Lawler.

INI 1
15:00 to 15:20 Afternoon Tea
15:20 to 16:20 Welding of the Backward SLE and Tip of the Forward SLE
Co-author: Steffen Rohde (University of Washington)

Let $\kappa\in(0,4]$. A backward chordal SLE$_\kappa$ process generates a conformal welding $\phi$, which is a random auto-homeomorphism of $\mathbb R$ that satisfies $\phi^{-1}=\phi$ and has a single fixed point: $0$. Using a stochastic coupling technique, we proved that the welding $\phi$ satisfies the following symmetry: Let $h(z)=-1/z$. Then $h\circ \phi\circ h$ has the same law as $\phi$. Combining this symmetry result with the forward/backward SLE symmetry and the conformal removability of forward SLE curve, we then derived some ergodic property of the tip of a forward SLE$_\kappa$ curve for $\kappa\in(0,4)$.

INI 1
16:20 to 16:30 Break
16:30 to 17:30 Boundary Measures and Natural Time Parameterization for SLE
Among Greg's many important contributions to the Schramm-Loewner evolution, he and his co-authors have in recent years been largely responsible for the construction and analysis of the natural time parameterization. The term ``natural'' refers to the parameterization favored by probabilists in the discrete curve setting, one unit of time per lattice site, and although it has not yet been shown that this discrete time parameterization converges it is still possible to define and study what should be the corresponding time parameterization for the continuum SLE curve. I will review many of the results in this direction that are due to Greg et al, along with some related results of my own in the boundary case, and then discuss some work in progress for extending the results on the boundary.
INI 1
Friday 19th June 2015
10:00 to 11:00 C Burdzy (University of Washington)
Twin peaks
Co-authors: Sara Billey (University of Washington), Soumik Pal (University of Washington), Lerna Pehlivan (University of Washington), Bruce Sagan (Michigan State University)

I will discuss some questions and results on random labelings of graphs conditioned on having a small number of peaks (local maxima). The main open question is to determine the distance between two peaks on a large discrete torus, assuming that the random labeling is conditioned on having exactly two peaks.

INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Loewner curvature
Co-author: Steffen Rohde (University of Washington)

Inspired by the geometric understanding of the SLE trace, there has been interest in studying how the deterministic Loewner equation encodes geometric properties of 2-dim sets into the 1-dim data of the driving function. Working in this vein, we define a new notion of curvature, called Loewner curvature, so-named because it captures key behavior of the trace curve of the Loewner equation. The Loewner curvature is defined for (nice enough) curves that begin at a marked boundary point of a Jordan domain and grow towards a second marked boundary point. We show that if this curvature is small, then the curve must remain a simple curve.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 From Internal DLA to self-interacting walks
I will present a few self-interacting random-walk-type models that somehow sit in between internal diffusion-limited aggregation and reinforced walks, associated to a strategy for proving a shape theorem for once-reinforced RW; and I will show some results in that direction.
INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons