2024-25

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

Abstract: Consider the following branching random walk model: individuals reproduce according to a branching process which is started from a single individual who is located at the origin. The children of a vertex receive random displacements away from the spatial location of the parent. The displacements away from a particular vertex may be dependent, with a distribution depending on the number of children. However, the set of displacements of a vertex’s children are independent of the sets of displacements associated with the children of other vertices.

We are interested in the setting where the genealogical tree is conditioned to have precisely n vertices and the offspring distribution is critical and of finite variance, so that the tree converges on rescaling distances by n^{-1/2} to Aldous’ Brownian continuum random tree. We also assume that the spatial displacements satisfy natural centring and finite variance conditions, so that, along a particular lineage, we observe something like a centred finite variance random walk. We investigate conditions under which the whole object then converges in distribution (in a suitable sense) to a Brownian motion indexed by the Brownian continuum random tree. (In order to do this, we actually make use of a standard tool known as a discrete snake, and prove convergence on rescaling to Le Gall’s Brownian snake driven by a Brownian excursion.)

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

Abstract: The Mallows distribution of parameter q is a family of distributions on the symmetric group generalizing the uniform distribution. Under the uniform distribution when $q=1$, the cycles of a random permutation are known to be approximately Poisson. I will discuss work establishing Gaussian behavior for the cycle structure of these permutations when q is not 1. The regime $q<1$ follows from known techniques, but the regime $q>1$ requires some new ideas. The proof uses the stationary Mallows process, an infinite stationary version of the Mallows permutation introduced by Gnedin and Olshanski. This is joint work with Tobias Müller and Teun Verstraaten.

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

Abstract: In this talk, we present the perturbative renormalization method by flow equations of the scalar field theory with a quartic self-interaction in position space in R4. This is mainly motivated by QFTs where translation invariance is broken, for instance QFTs in curved space-times or in spaces with boundaries. In these cases the momentum space representation is no more convenient. The idea of the proof is based on inductive bounds on the perturbative Schwinger distributions smeared with a suitable class of test functions, which imply tree decay between their position arguments. We present the trees and their associated weight factors and we explain how these combinatorial objects emerge naturally from the structure of the Polchinski flow equations together with the properties of the associated heat kernel.

Nov 27 - Giorgio Cipolloni (University of Arizona)

Title: Logarithmically correlated fields from non-Hermitian random matrices

Abstract: We study the Brownian evolution of large non-Hermitian matrices and show that their log-determinant converges to a 2+1-dimensional Gaussian field in the Edwards-Wilkinson regularity class, i.e. logarithmically correlated with respect to the parabolic distance. This gives a dynamical extension of the celebrated result by Rider and Virag (2006) proving that the fluctuations of the eigenvalues of Gaussian non-Hermitian matrices converge to a 2-dimensional log-correlated field. Our result holds out of equilibrium for general matrices with i.i.d. entries as an initial condition.

We also study the extremal values of these fields and demonstrate their logarithmic dependence on the matrix dimension.

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:

Abstract:

Seminars in Term 2.

Jan 8 -

Title:

Abstract:

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

Title:

Abstract:

Jan 22 - TBC

Title:

Abstract:

Jan 29 -

Title:

Abstract:

Feb 5 - James Norris (University of Cambridge)

Title:

Abstract:

Feb 12 -

Title:

Abstract:

Feb 19 - Benoit Dagallier (Université Paris Dauphine)

Title:

Abstract:

Feb 26 -

Title:

Abstract:

Mar 5 - Scott Armstrong (Sorbonne Université)

Title:

Abstract:

Mar 12 -

Title:

Abstract: