Organisers

Giuseppe CannizzaroLink opens in a new window and Nikolaos ZygourasLink opens in a new window

The probability seminar is held on Wednesdays 16-17 in B3.02.

Seminars in Term 1.

Oct 2 - Jon Keating (University of Oxford)

Title: Joint Moments

Abstract: I will talk about the joint moments of the characteristic polynomials of random unitary matrices and their derivatives, and will discuss briefly in this context the joint moments of the Riemann zeta-function and its derivates.

Oct 9 - Anna Ben-Hamou (LPSM Sorbonne Université)

Title: Cutoff for permuted Markov chains

Abstract: For a given finite Markov chain with uniform stationary distribution, and a given permutation on the state-space, we consider the Markov chain which alternates between random jumps according to the initial chain, and deterministic jumps according to the permutation. In this framework, Chatterjee and Diaconis (2020) showed that when the permutation satisfies some expansion condition with respect to the chain, then the mixing time is logarithmic in the size of the state space, and that this expansion condition is satisfied by almost all permutations. We will see that the mixing time can even be characterised much more precisely: for almost all permutations, the permuted chain has cutoff, at a time which only depends on the entropic rate of the initial chain.

Oct 16 - Christina Goldschmidt (University of Oxford)

Title: Trees and snakes

Our results improve on earlier theorems of various authors including Janson and Marckert for the case where the displacements are independent of the offspring numbers, and Marckert for the globally centred, global finite variance case restricted to bounded offspring distributions. Our proof of the convergence of the finite dimensional distributions makes essential use of a discrete line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin; the tightness proof adapts a method deployed by Haas and Miermont in the context of Markov branching trees.

This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.

Oct 23 - Joost Jooritsma (Oxford)

Title: Large deviations for the giant in spatial random graphs

Abstract: In (supercritical) Bernoulli bond percolation on Z^d the proportion of vertices in the largest cluster restricted to a volume-n box converges to θ: the probability that the origin lies in an infinite cluster. The probability that this proportion is smaller than θ-ε decays stretched exponentially with exponent strictly smaller than one. The probability that the largest cluster is much larger than expected decays exponentially. Thus, the upper tail decays much faster than the lower tail. In this talk, we will see that the discrepancy between the tails is reversed in supercritical spatial random graph models in which the degrees have heavy tails. In particular, we will focus on the soft heavy-tailed Poisson-Boolean model. The lower tail decays stretched exponentially, with an exponent that is determined by the strongest of three competing effects. However, the upper tail decays now polynomially, and thus decays much slower than the lower tail. We will give intuition for the exponent of this polynomial, which is determined by the generating function of the finite cluster-size distribution.
Joint work with Júlia Komjáthy and Dieter Mitsche.

Oct 30 - Jeffrey Kuan (Texas A&M University)

Title: Universality of dynamic processes using Drinfel'd twisters.

Abstract: The concept of 'universality' motivates a wide variety of probability and mathematical physics problems, going back to the classical central limit theorem. Most recently, the Kardar--Parisi--Zhang universality class has been proven to have Tracy--Widom fluctuations in the long-time asymptotics. In this talk, I will present a new universality result about the long-time asymptotics of so--called ``dynamic'' processes. The asymptotic fluctuations are related to the Tracy--Widom distribution. The proof will utilize a duality of Markov processes, which is constructed using Drinfel'd twisters of the quantum group $U_q(sl_2)$, viewed as a quasi--triangular quasi-Hopf algebra. The orthogonality of the duality functions allow for an asymptotic analysis.

Nov 6 - Jimmy He (Ohio)

Title: Cycles of Mallows permutations

Nov 13 - Ajay Chandra (Imperial College)

Title: Non-commutative singular SPDE

Abstract: In this talk, I will describe some recent progress in generalizing the theory of singular stochastic partial differential equations to treat equations in non-commutative probability theory - examples will include the stochastic quantization of Fermionic quantum field theories and also models involving free probability. No familiarity with non-commutative probability will be assumed.
This is based on joint and ongoing work with Martin Hairer and Martin Peev.

Nov 20 - Majdouline Borji (Max Planck Institute, Leipsig)

Title: Perturbative renormalization by flow equations in position space in d=4

Nov 27 - Giorgio Cipolloni (University of Arizona)

Title: Logarithmically correlated fields from non-Hermitian random matrices

Dec 4 - Tal Orenshtein (Università Milano-Bicocca)

Title: Random walks among random conductances as rough paths

Abstract: We shall discuss invariance principles for random walks among random conductances (RWRC) lifted to the p-variation rough paths space. Together with Jean-Dominique Deuschel and Nicolas Perkowski, we established in 2021 a rough path extension of the Kipnis-Varadhan theorem for additive functionals of Markov processes without requiring additional assumptions. This was then applied to the RWRC, albeit under restrictive conditions. This talk will focus on ongoing work with Johannes Bäumler, Noam Berger, and Martin Slowik, where we adopt a different approach by embedding the random walk, together with the environment process, into a space of shift-covariant functions. This framework is particularly natural for RWRC and has been highly effective in the classical case (i.e., without lifting the path). Within this setting, we establish an annealed invariance principle under weaker assumptions, as well as a quenched result for the random walk on the infinite cluster of a supercritical Bernoulli percolation in dimensions three and higher.

Dec 18 - SPECIAL SESSION - Pierre Patie (Cornell University)

Title: Novel algebraic insights into self-similarity

Abstract: Self-similarity is a pervasive concept in mathematics, appearing in fields as diverse as probability, operator theory, and mathematical physics. In this talk, I will begin with a brief overview of how self-similar structures enhance our understanding of scaling limits and universality. I will then present a novel and comprehensive perspective on the rich structure of self-similarity by exploring its connections with group representation theory and operator algebras. Particular attention will be given to the roles of the Bessel operator and the Laplacian, which reveal unexpected insights within this framework.

Seminars in Term 2.

Jan 8 - Hiroshi Kawabi (Keio University)

Title: Stochastic quantisation associated with the exp(φ)_{2}-quantum field model

Abstract: We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the exp(φ)_{2}-quantum field model or Hoegh-Krohn’s model. In this talk, we discuss the stochastic quantisation of this model driven by the space-time white noise. Combining key properties of Gaussian multiplicative chaos with a method for singular SPDEs, we construct a unique time-global solution to the responding parabolic stochastic quantisation equation in the full L^{1}-regime α^{2}<8π of the charge parameter α. We also identify the solution with an infinite dimensional diffusion process constructed by the Dirichlet form approach. The main part of this talk is based on joint work with Masato Hoshino (Osaka University) and Seiichiro Kusuoka (Kyoto University).

Jan 15 - Thierry Bodineau (IHES, Paris-Saclay) CANCELLED

Title: A renormalisation group perspective on functional inequalities

Abstract: Functional inequalities provide information on the structure of a probability measure and on the relaxation of associated stochastic dynamics to equilibrium. In this talk, we will describe a multiscale analysis for decomposing high-dimensional measures into simpler structures and derive from it functional inequalities. The strategy is based on the renormalization group method used in statistical physics to study the distribution of interacting particle systems. We will also review other related developments and in particular show that this decomposition of measures can be interpreted in terms of measure transport.

Jan 22 - Brett Koleshnik (University of Warwick)

Title: Sinai excursions

Abstract: Sinai (1992) initiated the study of random walks with persistently positive area processes. We find the precise asymptotic probability that the area process of a random walk bridge is an excursion. The asymptotics are related to subset counting formulas from additive number theory. Our results respond to a question of Caravenna and Deuschel (2008), which arose in the context of the pinning and wetting statistical physics models. Joint works with Michal Bassan (Oxford) and Serte Donderwinkel (Groningen) will be discussed.

Jan 29 - NO SEMINAR

Title:

Abstract:

Feb 5 - James Norris (University of Cambridge)

Title: Scaling limits for subcritical planar Laplacian growth models

Abstract : The framework of iterated random conformal maps allows to formulate and analyse
a range of planar random growth models, driven by harmonic measure on the boundary,
directly in the continuum, without an intermediate lattice model.
The talk will present some scaling limit theorems for these models in a certain
subcritical regime where local particle interactions can be controlled.
This is joint work with Vittoria Silvestri and Amanda Turner.

Feb 12 - Adam Harper (University of Warwick)

Title: Large fluctuations of random multiplicative functions

Feb 19 - Benoit Dagallier (Université Paris Dauphine)

Title: The Polchinski renormalisation group flow: an introduction

Abstract: I will introduce the Polchinski flow (or dynamics), a general framework to study asymptotic properties of statistical mechanics and field theory models, inspired by renormalisation group ideas.
The Polchinski dynamics has appeared recently under different names, such as stochastic localisation, and in very different contexts. Here I will motivate its construction in detail from a physics point of view and mention a few applications, such as connections with optimal transport. I will in particular explain how the Polchinski flow can be used to generalise Bakry and Emery’s Γ2 calculus to obtain functional inequalities which provide large-scale information on physics models. The talk is based on a review paper with Roland Bauerschmidt and Thierry Bodineau, accessible here: https://arxiv.org/pdf/2307.07619

Feb 26 - Luca Fresta (Universität Bonn)

Title: Density of States of Random Band Matrices close to the Edge of the Spectrum

Abstract: We study the random band matrix ensemble introduced by Disertori--Pinson--Spencer, characterised by the variance profile associated with the operator $(-W^{2}\Delta_{\mathbb{Z}^{3}}+1)^{-1}$, $\Delta_{\mathbb{Z}^{3}}$ being the Laplacian on the three-dimensional lattice and $W$ being the characteristic width of the band. For any energy $E$ close to the edge of the spectrum, namely for $1.8 <|E| <2$, we rigorously prove that the density of states follows Wigner’s semicircle law with power law corrections in $W^{-1}$, provided $W$ is sufficiently large,
depending on $E$. The proof relies on the supersymmetric approach of Disertori--Pinson--Spencer, and extends their result, which was previously established for energies $|E| \leq 1.8$. Joint work with M. Disertori.

Mar 5 - Scott Armstrong (Sorbonne Université)

Title: Homogenization, anomalous diffusion and multifractal fields

Abstract: I will discuss some recent joint works with Kuusi, Bou-Rabee and Vicol on the large-scale/ long-time behavior of diffusion processes advected by vector fields with many active length scales. The effect of the vector field can build up across many scales and effect the scaling of the variance of the process, which is called anomalous diffusion and is related to many physical phenomena such as turbulence. Certain particular models which we study were analyzed by physicists in the late 1980s, who used heuristic renormalization group arguments to make very precise predictions. Our work is about developing mathematical machinery that allow us to make these renormalization group arguments rigorous. This machinery is essentially a more sophisticated version of the analytic "coarse-graining" ideas developed in the last ten years in the context of quantitative homogenization for uniformly elliptic equations. Unlike in classical homogenization, in which there are essentially two asymptotically well-separated scales, here we need to deal with infinitely many scales which are not well-separated.

Mar 12 - Andreia Chapouto (Université de Versailles Saint-Quentin)

Title: Pathwise well-posedness of stochastic nonlinear dispersive equations with multiplicative noises

Abstract: Over the last decades, the well-posedness issue of stochastic dispersive PDEs with multiplicative noises has been extensively studied. However, this study was done primarily from the viewpoint of Ito solution theory, and pathwise well-posedness remained completely open. In this talk, I will present the first pathwise well-posedness results for stochastic nonlinear wave equations (SNLW) and stochastic nonlinear Schrödinger equations (SNLS) with multiplicative white-in-time/coloured-in-space noise. Here, we combine the operator-value controlled rough paths adapted to dispersive flows, together with random tensor estimates, and the Fourier restriction norm method adapted to controlled rough paths.