# Probability Seminar

*The probability seminar takes place on Wednesdays at 4:00 pm in room B3.02.*

Organisers: Wei Wu, Stefan Adams, Stefan Grosskinsky

###### Term 2 2020

**January 8: Marielle Simon (INRIA)**** CANCELLED**

**Title: **Hydrodynamic limit for an activated exclusion process

**January 15: Steffen Dereich (Münster)**

**Title: CLTs for stochastic gradient descent for stable manifolds**

**Abstract: **Nowadays stochastic gradient descent (SGD) algorithms are a standard tool for solving optimization problems. In this talk, we consider the case where the set of local minima is not discrete. This is, for instance, the case in deep learning with ReLU activation function where a single function is parametrized by a non-discrete set of parameters. In this talk we derive central limit theorems for Polyak-Ruppert averaged SGD where the set of local minima form an appropriate stable manifold. We recover the same rate of convergence as in the case of isolated attractors for step-sizes $\gamma_n=n^{-\gamma}$ with $\gamma\in(\frac34,1)$.

**January 22: no seminar**

Statistical aspects of geodesic flows in nonpositive curvature

**January 29: Perla Sousi (Cambridge)**

**Title: **TBA

**February 12: Ofer Busani (Bristol)
**

**Title:** TBA

**February 19: Harald Oberhauser (Oxford)**

**Title:** TBA

**February 26: Alex Watson (UCL)**

**Title:** TBA

**March 4: Franco Severo (IHES)**

**Title: **TBA

**March 11:** **Marielle Simon (INRIA)**

**Title: **Hydrodynamic limit for an activated exclusion process

**Abstract:** In this talk we will be interested in a one-dimensional exclusion process subject to strong kinetic constraints. More precisely, its stochastic short range interaction exhibits a continuous phase transition to an absorbing state at a critical value of the particle density. We will see that, in the active phase (i.e. for initial profiles smooth enough and uniformly larger than the critical density 1/2), the macroscopic behavior of this microscopic dynamics, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to the class of fast diffusion equations. The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

Based on a joint work with O. Blondel, C. Erignoux and M. Sasada

###### Term 1 2019-20

**October 2: Chiranjib Mukherjee (Münster)**

**Title: **Compactness, Large Deviations and a rigorous theory of the Polaron

**Abstract: **see pdf

**October 9: Benjamin Fehrman (Oxford)**

**Title**: Large deviations in interacting particle systems and stochastic PDE

**Abstract:**In this talk, which is based on joint work with Benjamin Gess, we will draw the link between large deviations in interacting particle systems and large deviations for certain classes of stochastic PDE. The motivating example will be the large deviations of the zero range process about its hydrodynamic limit. We will show informally that the corresponding rate function is identical to the rate function appearing for a degenerate stochastic PDE with nonlinear, conservative noise. Our primary result is a rigorous proof of this fact based upon an intricate treatment of the corresponding skeleton PDE, which is a nonlinear, energy-critical advection-diffusion equation.

**October 16: Hao Shen (Wisconsin)
**

**Title: **Stochastic Ricci flow on surfaces

**Abstract:** We introduce the Stochastic Ricci flow (SRF) in two spatial dimensions. It can be formally written in terms of the evolving Riemannian metric with a space-time noise which is “white” with respect to the metric; or in terms of the conformal factor, so that it becomes a natural quasi-linear generalization of the stochastic heat equation. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus. Modifications are needed for the SRF on general compact surfaces due to conformal anomaly. Under the flow, the total area of the surface follows a squared Bessel process. We also discuss some open questions.

**October 23: John Haslegrave (Warwick)**

**Title**: Phase transition and asymptotics for three-speed ballistic annihilation

**Abstract**: In the ballistic annihilation model, particles are emitted from a Poisson point process on the line, move at constant speed (chosen i.i.d. at initial time) and mutually annihilate when they collide. This model was introduced in the 1990s in physics, but once there are at least three possible speeds little is known rigorously about its behaviour. The most-studied discrete case has speeds of -1, 0 and +1, with symmetric probabilities. Here we prove that a phase transition takes place when stationary particles have probability 1/4, and give precise asymptotics for the decay of particles. This is joint work with Laurent Tournier and the late Vladas Sidoravicius.

**October 30: Tom Hutchcroft (Cambridge)
**

**Title:** Phase transitions in hyperbolic spaces

**Abstract:** Many questions in probability theory concern the way the geometry of a space influences the behaviour of random processes on that space, and in particular how the geometry of a space is affected by random perturbations. One of the simplest models of such a random perturbation is percolation, in which the edges of a graph are either deleted or retained independently at random with retention probability p. We are particularly interested in phase transitions, in which the geometry of the percolated subgraph undergoes a qualitative change as p is varied through some special value. Although percolation has traditionally been studied primarily in the context of Euclidean lattices, the behaviour of percolation in more exotic settings has recently attracted a great deal of attention. In this talk, I will discuss conjectures and results concerning percolation on the Cayley graphs of nonamenable groups and hyperbolic spaces, and give the main ideas behind our recent result that percolation in any transitive hyperbolic graph has a non-trivial phase in which there are infinitely many infinite clusters. The talk is intended to be accessible to a broad audience.

**November 6: David Belius (Basel)**

**Title:**The TAP-Plefka variational principle for mean field spin glasses

**Abstract:**The Thouless-Anderson-Palmer (TAP) approach to the Sherrington-Kirckpatrick mean field spin glass model was described in one of the earliest papers on this model, but has subsequently been a complementary rather than the central component in the theory that has emerged in theoretical physics and mathematics. In this this talk I will recall the TAP approach, and describe how it can be reinterpreted as a variational principle in the spirit of the Gibbs variational principle. Furthermore I will present a rigorous proof of this TAP-Plefka variational principle in the case of the spherical Sherrington-Kirkpatrick model, which allows to compute the free energy based purely on a TAP analysis.

**November 13: Jakob Björnberg (Gothenburg)**

**Title:** The interchange process with reversal

**Abstract:** The interchange process is a model for random permutations formed by composing random transpositions. Here we consider a variant of the interchange process where a fraction of the transpositions are replaced by `reversing transpositions'. A motivation for studying such processes is that they appear in the study of quantum models for magnetism, but they are also interesting in their own right. We will discuss a recent result obtained together with M. Kotowski, B. Lees and P. Milos which describes the scaling limit of the joint distribution of the largest cycles for the process defined on the complete graph.

**November 20: Alexey Bufetov (Bonn)**

**Title:** Color-position symmetry in interacting particle systems

**Abstract:** The asymmetric simple exclusion process (ASEP) is the evolution of a collection of particles on the integer lattice; particles interact according to simple rules and can be of various colors (equivalently, classes). In 2008 Amir-Angel-Valko established an interesting property of such processes: the color-position symmetry. We will discuss a generalization of this result and its new applications to the asymptotic behavior of this class of models.

**November 27: Alisa Knizel (Columbia)**

**Title:**Asymptotics of discrete β-corners processes via discrete loop equations

**Abstract:**We introduce and study stochastic particle ensembles which are natural discretizations of general β-corners processes. We prove that under technical assumptions on a general analytic potential the global fluctuations for the difference between two adjacent levels are asymptotically Gaussian. The covariance is universal and remarkably differs from its counterpart in random matrix theory. Our main tools are certain novel algebraic identities that we introduce. Based on joint work with Evgeni Dimitrov (Columbia University)

**December 4: No Seminar**