# 2022-23

##### Organisers

Giuseppe Cannizzaro and Vedran Sohinger

##### Seminars in Term 1.

**Oct 5** - Matija Vidmar, University of Ljubljana

* Title: *Noise Boolean algebras: classicality, blackness and spectral independence

*Abstract:*

**Oct 12** - Tommaso Rosati, University of Warwick

**Title:*** *Lyapunov exponents and global existence for SPDEs beyond order preservation.

** Abstract: **We present a new approach through a dynamic separation of scales to study Lyapunov exponents of multiplicative stochastic PDEs beyond the order preserving setting. We use related tools to establish global in time well-posedness for the stochastic Navier-Stokes equations with irregular noise and compare this to results for scalar conservation laws. Joint works with Martin Hairer and Ana Djurdjevac.

**Oct 19** - Rongfeng Sun, National University of Singapore

* Title: * A new correlation inequality for Ising models with external fields

** Abstract: **We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically 0. One corollary is that spin-spin correlation is maximised when the external field vanishes. In particular, the random field Ising model on Z^d, d ≥ 3, exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model on Z^d, d ≥ 3, satisfies the conjectured strong spatial mixing property in the entire high temperature regime. Based on joint work with Jian Ding and Jian Song.

**Oct 26** - Cristina Caraci, University of Zurich

* Title: *The excitation spectrum of two-dimensional Bose gases in the Gross-Pitaevskii regime

** Abstract: **I will discuss spectral properties of two dimensional Bose gases confined in a unit box with periodic boundary conditions. We assume that N particles interact through a repulsive two-body potential, with scattering length that is exponentially small in N, i.e. the Gross-Pitaevskii regime.

**Nov 2** - Alessandra Cipriani, University College London

* Title: *Properties of the gradient squared of the Gaussian free field

*Abstract: *

*jww Rajat Subhra Hazra (Leiden), Alan Rapoport (Utrecht) and Wioletta Ruszel (Utrecht)*

**Nov 9** - Kevin Yang, University of California, Berkeley

* Title: *Time-dependent KPZ equation from non-equilibrium Ginzburg-Landau SDEs

** Abstract: **This talk has two goals. The first is the derivation of a time-dependent KPZ equation (TDKPZ) from a time-inhomogeneous Ginzburg-Landau model. To our knowledge, said TDKPZ has not yet been derived from microscopic considerations. It has a nonlinear twist that is not seen in the usual KPZ equation, making it a more interesting SPDE.

The second goal is the universality of the method (for deriving TDKPZ), which should work beyond Ginzburg-Landau. In particular, we answer a question of deriving (TD)KPZ from asymmetric particle systems under natural fluctuation-scale versions of the assumptions in Yau’s relative entropy method and a log-Sobolev inequality. This gives some progress on open questions posed at a workshop on KPZ at the American Institute of Math. Time permitting, future directions (of both pure and applied mathematical flavors) will be discussed.

**Nov 16** - Sam Olesker-Taylor (University of Warwick)

* Title: *Random Walks on Random Cayley Graphs

** Abstract: **We investigate mixing properties of RWs on random Cayley graphs of a finite group G with k ≫ 1 independent, uniformly random generators, with 1 ≪ log k ≪ log |G|. Aldous and Diaconis (1985) conjectured that the RW on this random graph exhibits cutoff for any group G whenever k ≫ log |G| and further that the cutoff time depends only on k and |G|. It was established for Abelian groups. We disprove the second part of the conjecture by considering RWs on upper-triangular matrices. We extend this conjecture to 1 ≪ k ≲ log |G|, verifying a version of it for arbitrary Abelian groups under 'almost necessary' conditions on k. It is all joint work with Jonathan Hermon (now at UBC).

**Nov 23** - Pierre-François Rodriguez, Imperial College London

* Title: *Scaling in low-dimensional long-range percolation models

** Abstract: **The talk will present recent progress towards understanding the critical behavior of 3-dimensional percolation models exhibiting long-range correlations. The results rigorously exhibit the scaling behavior of various observables of interest and are consistent with scaling theory below the upper-critical dimension (expectedly equal to 6).

**Dec 1** - Sunil Chhita, University of Durham: **Seminar in MS.04 on Thursday, Dec. 1, 16-17.**

* Title: *Domino Shuffle and Matrix Refactorizations.

*Abstract:*

This talk is motivated by computing correlations for domino tilings of the Aztec diamond. It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. This model is of particular probabilistic interest due to being one of the few models having a boundary between polynomially and exponentially decaying macroscopic regions in the limit. One of the methods to compute correlations, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two- by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. In this talk, we present results to say that the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. This is based on joint work with Maurice Duits (Royal Institute of Technology KTH).

**Dec 7 **- Adrián Gonzáles Casanova, UNAM

**Title:*** *Sampling Duality

**Abstract:*** *Sampling Duality is stochastic duality using a duality function S(n,x) of the form ¨what is the probability that all the members of a sample of size n are of type -, given that the number (or frequency) of type - individuals is x¨. Implicitly this technique can be traced back to the work of Pascal. Explicitly it is studied in a paper of Martin Möhle in 1999. We will discuss several examples in which this technique is useful, including Haldane's formula and the long standing open question of the rate of the Muller Ratchet.

##### Seminars in Term 2.

**Jan 11 **- Trishen Gunaratnam, University of Geneva

* Title: * The tricritical point of the Blume-Capel model

** Abstract: **The Blume-Capel model is a ferromagnetic spin system where spins take values -1,0,+1. It can be thought of as an Ising model in an annealed random environment. It was introduced by Blume, and later studied by Capel, to capture phase transition in the absence of an external magnetic field. Despite its simplicity, the model is conjectured to have a surprisingly rich phase diagram. In particular, it is expected to exhibit a so-called tricritical point along its curve of critical points: a point which marks the boundary between continuous and discontinuous phase transition. In dimensions 2 and 3, the tricritical point is expected to be in a different universality classs to that of critical Ising. In this talk, I will describe results obtained in joint work with Dmitry Krachun and Christoforos Panagiotis where we show that at least one tricritical point exists in all dimensions.

**Jan 18: Note room change to B3.03 for this week only **- Alberto Chiarini, Università di Padova

* Title:* On the cost of covering a fraction of a macroscopic body by a simple random walk.

*Abstract:** * In this talk we aim at establishing large deviation estimates for the probability that a simple random walk on the Euclidean lattice (d>2) covers a substantial fraction of a macroscopic body. It turns out that, when such rare event happens, the random walk is locally well approximated by random interlacements with a specific intensity, which can be used as a pivotal tool to obtain precise exponential rates. Random interlacements have been introduced by Sznitman in 2007 in order to describe the local picture left by the trace of a random walk on a large discrete torus when it runs up to times proportional to the volume of the torus, and has been since a popular object of study. In the first part of the talk we introduce random interlacements and give a brief account of some results surrounding this object. In the second part of the talk we study the event that random interlacements cover a substantial fraction of a macroscopic body. This allows to obtain an upper bound on the probability of the corresponding event for the random walk. Finally, by constructing a near-optimal strategy for the random walk to cover a macroscopic body, we discuss a matching large deviation lower bound. The talk is based on ongoing work with M. Nitzschner (NYU Courant).

**Jan 25 **- Vittoria Silvestri, Università di Roma La Sapienza

**Title:**** **Explosive growth for a constrained Hastings–Levitov aggregation model

** Abstract: **The Hastings--Levitov (HL) growth models describe the formation of random aggregates in the complex plane via conformal maps. In this talk I will discuss a version of the HL models on the upper half plane, in which the growth is restricted to the cluster boundary. We will see that, although one might expect a shape theorem, this constrained model exhibits explosive behaviour, in that the cluster accumulates infinite diameter as soon as it reaches positive capacity. Based on joint work with Nathanael Berestycki.

**Feb 1 **- Seminar Postponed.

**Feb 8 **- Nikolay Barashkov, University of Helsinki

* Title: *Invariant measure for the Anderson wave equation

*Abstract:*

**Feb 15 **- **SEMINAR CANCELLED**

**Feb 22 **- Alessandra Occelli, Université d'Angers

* Title: *Nonlinear fluctuations of multi-species interacting particle systems

** Abstract: **We study the equilibrium fluctuations of an exclusion process evolving on the discrete ring with three species of particles, named A, B and C . We prove that proper choices of the density fluctuation fields (given by linear combinations of the fields associated to the conserved quantities t Publishhat match the prediction from mode coupling theory [Spohn 2014]) converge, in a suitable large scale limit, to stochastic partial differential equations, that can either be the Ornstein--Uhlenbeck equation or the stochastic Burgers equation. Based on a joint work with G. Cannizzaro, P. Gonçalves and R. Misturini.

**Mar 1 **- Serge Cohen, University of Toulouse

* Title: *Transition of the simple random walk on the graph of the ice-model

** Abstract: **The 6-vertex model is a seminal model for many domains in Mathematics and Physics. The sets of configurations of the 6-vertex model can be described as the set of paths in multigraphs. In this article the transition probability of the simple random walk on the multigraphs is computed. The unexpected point of the results is the use of continuous fractions to compute the transition probability.

**Mar 8 **- Natasha Blitvic, Queen Mary University London

* Title: *Combinatorial moment sequences

*Abstract:*

Take your favourite integer sequence. Is this sequence a sequence of moments of some probability measure on the real line? We will look at a number of interesting examples (some proven, others merely conjectured) of moment sequences in combinatorics. We will consider ways in which this positivity may be expected (or surprising!), the methods of proving it, and the consequences of having it. The problems we will consider will be very simple to formulate, but will take us up to the very edge of current knowledge in combinatorics, ‘classical’ probability, and noncommutative probability.

**Mar 15 **- Neil O'Connell, University College Dublin

* Title: *A (Toda-lly cool) Markov chain on reverse plane partitions

** Abstract: **I will discuss a natural Markov chain on reverse plane partitions which is closely related to the Toda lattice and has some remarkable properties. This talk is based on the paper https://arxiv.org/abs/2211.05718Link opens in a new window and aimed at a general probability audience. (The fun part of the title is borrowed from `Toda-lly cool stuff’, an informal and very accessible introduction to the Toda lattice, by Barbara Shipman, available at https://faculty.tcu.edu/richardson/seminars/barbaratoda.pdfLink opens in a new window.)

##### Seminars in Term 3. The talks are held in B3.02.

**Apr 26-** Eleanor Archer (Paris Nanterre University)

* Title:* Scaling limit of high-dimensional uniform spanning trees.

* Abstract:* A spanning tree of a finite connected graph G is a connected subgraph of G that touches every vertex and contains no cycles. In this talk we will consider uniformly drawn spanning trees of high-dimensional graphs, and show that, under appropriate rescaling, they converge in distribution as metric-measure spaces to Aldous’ Brownian CRT. This extends an earlier result of Peres and Revelle (2004) who previously showed a form of finite-dimensional convergence. Based on joint works with Asaf Nachmias and Matan Shalev.

**May 3-** Benjamin Fehrman (University of Oxford)

* Title: * Non-equilibrium fluctuations and parabolic-hyperbolic PDE with irregular drift

**Abstract:*** * Non-equilibrium behavior in physical systems is widespread. A statistical description of these events is provided by macroscopic fluctuation theory, a framework for non-equilibrium statistical mechanics that postulates a formula for the probability of a space-time fluctuation based on the constitutive equations of the system. This formula is formally obtained via a zero noise large deviations principle for the associated fluctuating hydrodynamics, which postulates a conservative, singular stochastic PDE to describe the system far-from-equilibrium. In this talk, we will focus particularly on the fluctuations of the zero range process about its hydrodynamic limit. We will show how the associated MFT and fluctuating hydrodynamics lead to a class of conservative SPDEs with irregular coefficients, and how the study of large deviations principles for the particles processes and SPDEs leads to the analysis of parabolic-hyperbolic PDEs in energy critical spaces. The analysis makes rigorous the connection between MFT and fluctuating hydrodynamics in this setting, and provides a positive answer to a long-standing open problem for the large deviations of the zero range process.

**May 10- **Leonardo Tolomeo (University of Edinburgh)

* Title:* Statistical mechanics of the focusing nonlinear Schrödinger equation

** Abstract: **In this talk, we will complete the program on the (non-)construction of the Gibbs measures for focusing nonlinear Schrödinger equations.

We will review the classical results of Lebowitz-Rose-Speer and Brydges-Slade via more modern techniques, and explain the source of the new phase transition in dimension 3.

**May 17-** Thomas Bothner (University of Bristol)

* Title: *Bulk spacings in non-Hermitian matrix models

* Abstract:* Random matrix eigenvalue spacings tend to show up in problems not directly related to random matrices: for instance, bumper to bumper distances of parked cars in a number of roads in central London are well represented by the so-called eigenvalue bulk spacing distribution of a suitable Hermitian matrix model. In this talk we will first survey several occurrences of these Hermitian spacing distributions and afterwards try to generalise them to non-Hermitian models. As it turns out, the theory of integrable systems, especially Painlevé special function theory, plays a crucial role in this field. Based on arXiv:2212.00525, joint work with Alex Little (Bristol).

**May 24- **Fabio Toninelli (Technische Universität Wien)

* Title:* Mixing time of random tilings

*I will discuss rhombus tilings of a (tilable) finite subset D of the plane and Markov chains (Glauber dynamics) that are reversible with respect to the uniform measure over all possible tilings. What is the mixing time T_{mix}? Under some natural conditions on the domain D, it is expected that T_{mix} grows like L^{2+o(1)} in continuous time, where L is the diameter of D. I will discuss recent and less recent results in this direction, as well as some intriguing open problems (based on joint works with Benoit Laslier).*

**Abstract:****May 31- Seminar cancelled**

**June 7- **Augusto Texeira (IMPA)

* Title:* Fluctuation bounds for Random Walks on Dynamic Environments via Russo-Seymour-Welsh

*In this talk we will review a few general statements about Random Walks on Dynamic Random Environments (RWDRE). As expected, if the random environment satisfies stronger mixing conditions, more can be said about the behavior of the random walk. Then we will focus on a recent work establishing lower bounds on fluctuations for certain RWDRE models. This result requires the environment to be symmetric, satisfy the FKG inequality as well as a mixing condition. The techniques employed in the proof are inspired by percolation theory and the Russo-Seymour-Welsh (RSW) inequality. We finally exemplify our main result with two examples of environments: a class of Gaussian fields and a Confetti percolation medium.*

**Abstract:****June 14- **Willem van Zuijlen (Freie Universität Berlin and WIAS)

* Title: *Weakly self avoiding walk in a random potential

* Abstract:* We investigate a model of simple-random walk paths in a random environment that has two competing features: an attractive one towards the highest values of a random potential, and a self-repellent one in the spirit of the well-known weakly self-avoiding random walk. We tune the strength of the second effect such that they both contribute on the same scale as the time variable tends to infinity. In this talk I will discuss our results on the identification of (1) the logarithmic asymptotics of the partition function, and (2) of the path behaviour that gives the overwhelming contribution to the partition function. This is joint work with Wolfgang König, Nicolas Pétrélis and Renato Soares dos Santos.

**June 21- **Daniel Kious (University of Bath)

**June 28-(Room change; seminar takes place in B3.03) **Amanda Turner (University of Leeds)