# 2020-21

##### Organisers

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

Zoom link: https://zoom.us/j/93299847543

##### Online Format

This year, the probability seminar will take place online in collaboration with Queen Mary and Bristol. During Term 1 the seminar will run on **Fridays, 3-4pm** via Zoom. The seminar will be followed by an online social gathering with the speaker.

##### Schedule for Term 1

(also available at Bristol and Queen Mary webpages)

**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 30:** Perla Sousi (Cambridge)

**Title:** TBA

**Abstract:** TBA

**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.

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

**Title:** TBA

**Abstract:** TBA

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

**Title:** TBA

**Abstract:** TBA

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

**Title:** TBA

**Abstract:** TBA

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

**Title:** TBA

**Abstract:** TBA

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

**Title:** TBA

**Abstract:** TBA

**October 16:** Firas Rassoul-Agha (Utah) [Zoom link]

**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.