# Probability Seminar

##### Organisers

Leonardo T. Rolla and Nikos Zygouras in collaboration with Sasha Sodin (Queen Mary), Ofer Busani and Benjamin Lees (Bristol).

##### Upcoming seminars

For the Zoom link, please contact the organisers the day before.

**January 20** 16:00 - John Sylvester (University of Glasgow)

**Title:** Multiple Random Walks on Graphs: Mixing Few to Cover Many

**Abstract: **In this talk we will consider k random walks that are run independently and in parallel on a finite, undirected and connected graph. Alon et al. (2008) showed that the effect of increasing k, the number of walkers, does not necessarily effect the cover time (time until each vertex has been visited by at least one walk) in a straightforward manner. Despite subsequent progress in the area, the problem of finding a general characterisation of multiple cover times for worst-case start vertices remains an open problem.

We shall present some of our recent results concerning the multiple cover time from independent stationary starting vertices. Firstly, we improve and tighten various bounds, which allow us to establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Secondly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

This is joint work with Nicolás Rivera and Thomas Sauerwald https://arxiv.org/abs/2011.07893

**January 27** 16:00 - Alexander Dunlap (Courant Institute of Mathematical Sciences)

**Title:**A forward-backward SDE from the 2D nonlinear stochastic heat equation

**Abstract:**I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover the log-normal behavior proved by Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).

**February 03** 16:00 - Jess Jay (University of Bristol)

**Title:** TBA

**Abstract:** TBA

**February 10** 16:00 - Federico Camia (NYU-Abu Dhabi)

**Title:** TBA

**Abstract:** TBA

**February 17** 16:00 - Davide Gabrielli (University of L'Aquila)

**Title:** TBA

**Abstract:** TBA

**February 24** 16:00 - Gady Kozma (Weizmann Institute of Science)

**March 03** 16:00 - Jon Keating (University of Oxford)

**March 10** 16:00 - Axel Saenz Rodriguez (University of Warwick)

**March 17** 16:00 - Vittoria Silvestri (University of Rome La Sapienza)

##### Past seminars

**January 13** 16:00 - Bálint Virág (University of Toronto)

**Title:** The heat and the landscape

**Abstract:** If lengths 1 and 2 are assigned randomly to each edge in the planar grid, what are the fluctuations of distances between far away points?This problem is open, yet we know, in great detail, what to expect. The directed landscape, a universal random plane geometry, provides the answer to such questions. In some models, such as directed polymers, the stochastic heat equation, or the KPZ equation, random plane geometry hides in the background. Principal component analysis, a fundamental statistical method, comes to the rescue: BBP statistics can be used to show that these models converge to the directed landscape.

**December 11:** Herbert Spohn (Munich)

**Title:** Generalized Gibbs measures of the Toda lattice

**Abstract:** According to statistical mechanics, Gibbs measures for a many-particle system are constructed from number, momentum, and energy, which are believed to be generically the only locally conserved fields. The classical Toda lattice is an integrable system and thus possesses a countable list of local conservation laws. Accordingly Gibbs measures of the Toda chain are indexed by an infinite set of parameters, rather than only three. This is the meaning of “generalised" in the title. Specifically for the Toda lattice, we will discuss the structure of generalised Gibbs measures and point out the connection to the one-dimensional log gas. This information is a central building block when writing down the appropriate Euler type equations for the Toda lattice.

**December 4:** Kieran Ryan (Queen Mary)

**Title:** The quantum Heisenberg XXZ model and the Brauer algebra

**Abstract:** Several quantum spin systems have probabilistic representations as interchange processes. Some of these processes can be thought of as continuous time random walks on the symmetric group, which in turn has led to study using representation theory. We will in particular discuss the spin-½ quantum Heisenberg XXZ model (and analogous models in higher spins, which are not the XXZ). This model, in a similar way to study using the symmetric group for other models, can be studied using the Brauer algebra. We will introduce this algebra, and its representation theory. Using this we obtain the free energy of the system when the underlying graph is the complete graph, which further lets us determine phase diagrams.

**November 27**: Horatio Boedihardjo (Warwick)

**Title:** The signature of a path

**Abstract: **There has been a lot of activities recently pingrov path-wise results for stochastic differential equations. Some of these activities has been inspired by rough path theory which enables integration against the irregular sample paths to be defined in a path-wise, as opposed to $L^2$, sense. While there have been many existence and uniqueness results, a natural next step is to develop descriptive path-wise theory for differential equations driven by specific path. In this talk, we study a particular example of differential equation and its solution, known as the signature. It is one of the simplest equations studied in rough path theory and has some interesting properties that general equations do not have.

**November 20:** Michael Damron (Georgia Tech)

**Title:** Critical first-passage percolation in two dimensions

**Abstract:** In 2d first-passage percolation (FPP), we place nonnegative i.i.d. weights $(t_e)$ on the edges of $Z^2$ and study the induced weighted graph pseudometric $T = T(x,y)$. If we denote by $p = P(t_e = 0)$, then there is a transition in the large-scale behavior of the model as $p$ varies from $0$ to $1$. When $p < 1/2, T(0,x)$ grows linearly in $x$, and when $p > 1/2$, it is stochastically bounded. The critical case, where $p = 1/2$, is more subtle, and the sublinear growth of $T(0,x)$ depends on the behavior of the distribution function of $t_e$ near zero. I will discuss my work over the past few years that (a) determines the exact rate of growth of $T(0,x)$, (b) determines the "time constant" for the site-FPP model on the triangular lattice and, more recently (c) studies the growth of $T(0,x)$ in a dynamical version of the model, where weights are resampled according to independent exponential clocks. These are joint works with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.

**November 13:** Ewain Gwynne (Chicago)

**Title:** Tightness of supercritical Liouville first passage percolation

**Abstract:** Liouville first passage percolation (LFPP) with parameter $\xi >0$ is the family of random distance functions on the plane obtained by integrating $e^{\xi h_\epsilon}$ along paths, where $h_\epsilon$ for $\epsilon >0$ is a smooth mollification of the planar Gaussian free field. Previous work by Ding-Dub\'edat-Dunlap-Falconet and Gwynne-Miller showed that there is a critical value $\xi_{\mathrm{crit}} > 0$ such that for $\xi < \xi_{\mathrm{crit}}$, LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric: the so-called $\gamma$-Liouville quantum gravity metric for $\gamma = \gamma(\xi)\in (0,2)$. Recently, Jian Ding and I showed that LFPP admits non-trivial subsequential scaling limits for all $\xi > 0$. For $\xi >\xi_{\mathrm{crit}}$, the subsequential limiting metrics do \emph{not} induce the Euclidean topology. Rather, there is an uncountable, dense, Lebesgue measure-zero set of points $z\in\mathbb C $ such that $D_h(z,w) = \infty$ for every $w\in\mathbb C\setminus \{z\}$. We expect that the subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$. I will discuss the properties of the subsequential limiting metrics, their connection to Liouville quantum gravity, and several open problems.

**November 6:** Renan Gross (Weizmann)

**Title:** Stochastic processes for Boolean profit

**Abstract:** Not even influence inequalities for Boolean functions can escape the long arm of stochastic processes. I will present a (relatively) natural stochastic process which turns Boolean functions and their derivatives into jump-process martingales. There is much to profit from analyzing the individual paths of these processes: Using stopping times and level inequalities, we will reprove an inequality of Talagrand relating edge boundaries and the influences, and say something about functions which almost saturate the inequality. The technique (mostly) bypasses hypercontractivity. Work with Ronen Eldan. For a short, animated video about the technique (proving a different result, don't worry), see here: https://www.youtube.com/watch?v=vPLHAt_iv-0.

**October 30:** Perla Sousi (Cambridge)

**Title:** The uniform spanning tree in 4 dimensions

**Abstract:** A uniform spanning tree of ℤ⁴ can be thought of as the "uniform measure" on trees of ℤ⁴. The past of 0 in the uniform spanning tree is the finite component that is disconnected from infinity when 0 is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length n, that it has volume at least n and that it reaches the boundary of the box of side length n around 0. Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits "mean-field" critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is joint work with Tom Hutchcroft.

**October 23: **Inés Armendáriz (Buenos Aires)

**Title:** Gaussian random permutations and the boson point process

**Abstract:** We construct an infinite volume spatial random permutation associated to a Gaussian Hamiltonian, which is parametrized by the point density and the temperature. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Bose-Einstein condensation occurs for dimensions 3 or larger, above a critical density, and is manifest in this representation by the presence of cycles of macroscopic length. For subcritical densities we define the spatial random permutation as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For dimensions greater than or equal to 3 and supercritical densities, we define the spatial permutation as the superposition of independent realizations of the Gaussian loop soup at critical density and Gaussian random interlacements at the remaining density. We show some properties of these spatial permutations, in particular that the point marginal is the boson point process, for any point density.This is joint work with P.A. Ferrari and S. Yuhjtman.

**October 16:** Firas Rassoul-Agha (Utah)

**Title:** Geometry of geodesics through Busemann measures in directed last-passage percolation

**Abstract:** We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and describe geometry properties of the full set of semi-infinite geodesics in a typical realization of the random environment. The main tool is the Busemann functions viewed as a stochastic process indexed by the asymptotic direction. In the exactly solvable exponential model we give a complete characterization of the uniqueness and coalescence structure of the entire family of semi-infinite geodesics. Part of our results concerns the existence of exceptional (random) directions in which new interesting instability structures occur. This is joint work with Christopher Janjigian and Timo Seppalainen.

**October 8 (notice special day and time- Thursday, 2pm): **Naomi Feldheim (Bar Ilan)

**Title:** Persistence of Gaussian stationary processes

**Abstract: **Let *f* : ℝ→ℝ be a Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution. What is the probability that *f* remains above a certain fixed line for a long period of time? This simple question, which was posed by mathematicians and engineers more than 60 years ago (e.g. Rice, Slepian), has some surprising answers which were discovered only recently. I will describe how a spectral point of view leads to those results. Based on joint works with O. Feldheim, F. Nazarov, S. Nitzan, B. Jaye and S. Mukherjee.