# Seminars & Events for Topics in Probability

##### The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

We discuss the proof that the Tutte embeddings (a.k.a. harmonic or barycentric embeddings) of certain random planar maps converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and $\gamma$ ranges from $0$ to $2$ as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLE$_{\kappa}$ for $\kappa =16/\gamma^2$ (in the annealed sense) and that the embedded random walk converges to Brownian motion (in the quenched sense).

##### Convergence of finite-range weakly asymmetric exclusion processes on a circle

We consider spatially periodic growth models built from weakly asymmetric exclusion processes with finite-range jumps and rates depending locally on configuration. We prove that at a large scale and after renormalization these processes converge to the Hopf-Cole solution of the KPZ equation driven by Gaussian space-time white noise. Since the driving noise of the discrete equation decorrelates slowly in time, the hydrodynamic behaviour of the system needs to be exploited.

##### Progress in showing cutoff for random walks on the symmetric group

Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed state. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group.

##### Convergence of percolation-decorated triangulations to SLE and LQG

The Schramm-Loewner evolution (SLE) is a family of random fractal curves, which is the proven or conjectured scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Liouville quantum gravity (LQG) is a model for a random surface which is the proven or conjectured scaling limit of discrete surfaces known as random planar maps (RPM). We prove that a percolation-decorated RPM converges in law to SLE-decorated LQG in a certain topology. This is joint work with Bernardi and Sun. We then discuss works in progress with the goal of strengthening the topology of convergence of RPM to LQG by considering conformal embeddings of the RPM into the complex plane. This is joint with Garban, Gwynne, Miller, Sepulveda, Sheffield, and Sun.

##### Comparing exponential and Erdős–Rényi random graphs, and a general bound on the distance between Bernoulli random vectors

We present a general bound on the Wasserstein distance between two arbitrary distributions of sequences of Bernoulli random variables. The bound is in terms of 1) a mixing quantity for the Glauber dynamics of one of the sequences, and 2) a simple expectation of the other. The result is applied to estimate, with explicit error, expectations of functions of random vectors for some Ising models and exponential random graphs in "high temperature" regimes. Joint work with Gesine Reinert.