# Probability Seminar

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

Organisers: Hendrik Weber, Nikos Zygouras

###### Term 1 2017-18

4 October: *Aran Raoufi* (IHES)

**Title:** Translation invariant Gibbs states of Ising model

**Abstract:** We prove that for any amenable graph, for the ferromagnetic Ising model at any inverse temperature $\beta$, every automorphism invariant Gibbs state is a linear combination of the pure plus and minus states.

11 October: Kurt Johansson (KTH)

**Title:** Two-time distribution in last-passage percolation

**Abstract:** I will discuss a new approach to computing the two-time distribution in last-passage percolation with geometric weights. This can be interpreted as the correlations of the height function at a spatial point at two different times in the equivalent interpretation as a discrete polynuclear growth model. I will also discuss the problem of multiple spatial points at the two times. The new approach is closer to standard random matrix theory (or determinantal point process) computations compared the one in my paper "Two time distribution in Brownian directed percolation", Comm. Math. Phys. 351 (2017)

18 October: Wei Wu (Warwick)

**Title: ** Extremal and local statistics for gradient field models

**Abstract: **We study the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field) in two dimension. These log-correlated non-Gaussian random fields arise in different branches of statistical mechanics. Existing results were mainly focused on the CLT for the linear functionals. In this talk I will describe some recent progress on the global maximum and local CLT for the field, thus confirming they are in the Gaussian universality class in a very strong sense. The proof uses a random walk representation (a la Helffer-Sjostrand) and an approximate harmonic coupling (by J. Miller).

25 October: *Scott Smith* (Leipzig)

**Title**: Quasi-linear parabolic PDE's with singular inputs

**Abstract: **The present talk is concerned with quasi-linear parabolic equations which are ill-posed in the classical distributional sense. In the semi-linear context, the theory of regularity structures provides a solution theory which applies to a large class of equations with suitably randomized inputs. Recently, Otto and Weber have initiated an approach to the quasi-linear setting and developed new tools for the so-called reconstruction and integration steps (in the language of regularity structures). We will discuss recent efforts to extend their tools to treat more singular noises. Towards this end, we introduce a new interpretation of their approach in terms of a suitable notion of modelled distribution and present several generalizations of their tools. This is a joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.

1 November: *Dan Betea* (Paris)

**Title**: The Pfaffian free boundary Schur process and asymptotics

**Abstract**: We introduce the machinery of Schur processes in general and in particular of such processes when one or both boundaries are free. These latter processes turn out to have explicitly computable pfaffian correlations amenable to asymptotic analysis. Applications include KPZ-type asymptotics of last passage percolation models rediscovering results of Baik--Rains, of symmetric plane partitions, of plane overpartitions (a form of domino tilings) and of others yet to be investigated. The machinery behind the results is the integrable free fermionic machinery of the Kyoto school and a (possibly) new type of Wick lemma, similar (spiritually) to the Wick lemma in finite temperature recently employed by Le Doussal, Majumdar and Schehr in the case of periodic boundary conditions and and by Johansson in his GUE-in-finite-temperature model. Joint work with Jeremie Bouttier (ENS Lyon and CEA), Peter Nejjar (IST Vienna) and Mirjana Vuletic (UMass Boston).

8 November: Lorenzo Taggi (Darmstadt)

**Title**: Self-avoiding random walk interacting with an ensemble of self-avoiding polygons

**Abstract**: We consider a self-avoiding walk connecting two opposite sides of a box in $Z^d$ and interacting with an ensemble of self-avoiding polygons by mutual exclusion. The probability of a realization of this process is proportional to $x^E$, where $E$ is the total number of edges in the box and $x>0$ is a parameter. A central question for this model is what is the behavior of such a random walk as the side length L of the box goes to infinity. Our main result is that, when x is small enough, the fluctuations of the walk are at most of order $(\log L)^3 \sqrt{L}$. In this regime one expects the walk to converge to Brownian motion under diffusive scaling. Our result is obtained by employing a renewal argument (Ornstein-Zernike method) which requires a coupling with a sub-critical Galton-Watson process and a proof that the polygon length admits uniformly bounded exponential moments. The content of this talk is based on a joint work with Volker Betz.

15 November: Vedran Sohinger (Warwick)

**Title**: A microscopic derivation of time-dependent correlation functions of the 1D nonlinear Schrödinger equation

**Abstract:** The nonlinear Schrödinger equation (NLS) is a nonlinear PDE which admits an invariant Gibbs measure. The construction of these measures was given in the constructive field theory literature in the 1970s and their invariance was first rigorously proved by Bourgain in the 1990s. Since then, Gibbs measures have become an important tool in constructing solutions for low regularity random initial data.

The NLS can also be viewed as a classical limit of many-body quantum dynamics. In this context, it is natural to ask how one can obtain the Gibbs measure as a limit of many-body quantum Gibbs states. In the first part of the talk, I will review some results on this problem, obtained in earlier joint work with J. Fröhlich, A. Knowles, and B. Schlein. The main part of the talk is devoted to the time-dependent problem. I will explain how to derive time-dependent correlation functions of the NLS in a limit from corresponding quantum objects in one dimension. This result holds for nonlocal interactions with bounded convolution potential. I will also explain how one can obtain a partial result for local interactions on the circle. This is joint work with J. Fröhlich, A. Knowles, and B. Schlein.

22 November: Dominik Schröder (IST Austria)

**Title:** Random matrices with slow correlation decay

**Abstract:** We first present a short introduction to random matrix theory and its motivations from quantum physics. In the main part of the talk we review some recent results on the local eigenvalue statistics of various random matrix models generalising the classical Wigner random matrices with independent identically distributed zero mean entries. We demonstrate that the celebrated Wigner-Dyson-Mehta universality conjecture also extends to correlated random matrices with a finite polynomial decay of correlations and arbitrary expectation. Our proof relies on a quantitative stability analysis of the matrix Dyson equation (MDE) as well as on a systematic diagrammatic control of a multivariate cumulant expansion.

29 November: Nathanaël Berestycki (Cambridge)

**Title:** A characterisation of the Gaussian free field

**Abstract:** Over the last decades, the planar Gaussian free field has been conjectured, and in some cases proved, to arise as a universal scaling limit for a broad range of models from statistical physics. In this talk we prove that any random distribution satisfying conformal invariance, a form of domain Markov property and a fourth moment condition must be a multiple of the Gaussian free field. We will also discuss several open problems concerning the situation beyond the Gaussian case.

Joint work (in progress) with Ellen Powell and Gourab Ray.

6 December: Andreas Kyprianou (Bath)

**Title:** Terrorists never congregate in even numbers

**Abstract:** We analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non-random in the large-scale limit. Moreover, we discover that, in the limit of small fragmentation rate, these processes exhibit a universal heavy tailed distribution with exponent 3/2. In addition, we observe a strange phenomenon that if coalescence of clusters always involves 3 or more blocks, then the large-scale limit has no even sided blocks. Some complementary results are also presented for exchangeable fragmentation-coalescence processes on partitions of natural numbers. In this case one may work directly with the infinite system and we ask whether the process can come down from infinity. The answer reveals a remarkable dichotomy. This is based on two different pieces of work with Tim Rogers, Steven Pagett and Jason Schweinsberg

**Term 2, 2017-18**

10 January: Balint Toth (Bristol)

**Title: **Quenched CLT for random walk in doubly stochastic random environment

**Abstract: **I will present the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition. The proof relies on non-trivial extension of Nash's moment bound to this context and on down-to-earth concrete functional analytic arguments.

17 January: Benjamin Gess (Leipzig)

**Title:** Path-by-path regularization by noise for scalar conservation laws

**Abstract:** We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [G., Souganidis; Comm. Pure Appl. Math. (2017)]. We show that $(\rho,\gamma)$-irregularity is a sufficient path-by-path condition implying improved regularity. In addition, we introduce a new path-by-path scaling property which is also shown to be sufficient to imply regularizing effects.

24 January: Elisabetta Candellero (Warwick)

**Title:** Coexistence of competing first-passage percolation on hyperbolic graphs.

**Abstract**: We consider two first-passage percolation processes $FPP_1$ and $FPP_{\lambda}$, spreading with rates $1$ and $\lambda > 0$ respectively, on a non-amenable hyperbolic graph G with bounded degree. $FPP_1$ starts from a single source at the origin of $G$, while the initial con figuration of $FPP_{\lambda}$ consists of countably many seeds distributed according to a product of iid Bernoulli random variables of parameter $\mu > 0$ on $V (G)\setminus\{o\}$. Seeds start spreading $FPP_{\lambda}$ after they are reached by either $FPP_1$ or $FPP_{\lambda}$. We show that for any such graph $G$, and any fixed value of $\lambda > 0$ there is a value $\mu_0 = \mu_0(G,\lambda ) > 0$ such that for all $0 < \mu < \mu_0$ the two processes coexist with positive probability. This shows a fundamental difference with the behavior of such processes on $Z^d$. (Joint with Alexandre Stauffer.)

31 January: Wei Qian (Cambridge)

**Title**: On loop soups, GFF, and the coupling of Dirichlet and Neumann GFFs via commun level-lines

**Abstract**: We discuss a simultaneous coupling between the critical loop soup, GFF and CLE4. We point out a simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary touching 0-level lines (these signs are alternating for the zero-boundary GFF) in order to obtain a free boundary GFF. Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are also discussed. It is joint work with Wendelin Werner.

7 February: Marton Balazs (Bristol)

**Title: **Jacobi triple product via the exclusion process"

**Abstract: **I will give a brief overview of very simple, hence maybe less investigated

structures in interacting particle systems: reversible product blocking

measures. These turn out to be more general than most people would think, in

particular asymmetric simple exclusion and nearest-neighbour asymmetric zero

range processes both enjoy them. But a careful look reveals that these two are

really the same process. Exploitation of this fact will give rise to the Jacobi

triple product formula - an identity previously known from number theory and

combinatorics. I will derive it from pure probability this time, and I hope to

surprise my audience as much as we got surprised when this identity first

popped up in our notebooks.

14 February Ilya Chevyrev (Oxford)

**Title**: Renormalising SPDEs in regularity structures

**Abstract**: Recent work in regularity structures has provided a robust solution theory for a wide class of singular SPDEs. While much progress has been made on understanding the analytic and algebraic aspects of renormalisation of the driving signal, until recently the action of renormalisation on the equation still needed to be performed by hand. In this talk, I will give a systematic description of the renormalisation procedure directly on the level of the PDE, thus providing a general black-box local existence and stability result of a large class of SPDEs. Joint work with Y. Bruned, A. Chandra, and M. Hairer.

21 February: Antti Knowles (Geneva)

**Title:** Mesoscopic eigenvalue correlations of random matrices.

**Abstract:** Ever since the pioneering works of Wigner, Gaudin, Dyson, and Mehta, the correlations of eigenvalues of large random matrices on short scales have been a central topic in random matrix theory. On the microscopic spectral scale, comparable with the typical eigenvalue spacing, these correlations are now well understood for Wigner matrices thanks to the recent solution of the Wigner-Gaudin-Dyson-Mehta universality conjecture. In this talk I focus on eigenvalue density-density correlations between eigenvalues whose separation is much larger than the microscopic spectral scale; here the correlations are much weaker than on the microscopic scale. I discuss to what extent the Wigner-Gaudin-Dyson-Mehta universality remains valid on such larger scales, for Wigner matrices and, if time allows, random band matrices.

28 February: Hugo Duminil-Copin (IHES)

**Title**: Emergent Planarity in two-dimensional Ising Models with finite-range Interactions

**Abstract**: In this talk, we revisit the known Pfaffian structure of the boundary spin correlations, which is given a new explanation through simple topological considerations within the model's random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on $\mathbb Z^2$ with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.

7 March: Julian Fischer (IST Austria)

**Title:** TBA

**Abstract:** TBA

14 March: Marcin Lis (Cambridge)

**Title**: TBA

**Abstract**: TBA

**Term 3, 2017-2018**

25 April: Jan Maas (IST, Vienna)

**Title**: TBA

**Abstract**: TBA

2 May: Yvan Velenik (Geneva)

**Title:** TBA

**Abstract:** TBA

9 May: Marco Furlan (Paris)

**Title:** TBA

**Abstract:** TBA

16 May: Balint Virag (Toronto)

**Title**: TBA

**Abstract**: TBA

23 May: Patricia Goncalves (Lisbon)

**Title:** TBA

**Abstract:** TBA

30 May: Nicolas Perkowski (Humboldt)

**Title:** TBA

**Abstract**: TBA

6 June: Minmin Wang (Bath)

**Title:** TBA

**Abstract:** TBA

27June: Xue-Mei Li (Imperial)

**Title:** TBA

**Abstract:** TBA