# 2022-23

##### Organisers

Giuseppe Cannizzaro and Vedran Sohinger

##### Seminars in Term 1. The seminars are held in B3.03, Wednesday 16-17.

**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

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**Jan 18 **- Alberto Chiarini, Università di Padova

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

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**Feb 1 **- Bálint Tóth, University of Bristol

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

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**Feb 15 **- Ellen Powell, University of Durham

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

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

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

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

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