# 2017-18

*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:** TBA

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17 January: Benjamin Gess (Leipzig)

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24 January: Elisabetta Candellero (Warwick)

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31 January: Wei Qian (Cambridge)

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7 February: Marton Balazs (Bristol)

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14 February: Peter Friz (TU Berlin)

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21 February: Antti Knowles (Geneva)

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28 February: Hugo Duminil-Copin (IHES)

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7 March: Julian Fischer (IST Austria)

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14 March: Marcin Lis (Cambridge)

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**Term 3, 2017-2018**

25 April: Jan Maas (IST, Vienna)

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2 May: Yvan Velenik (Geneva)

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9 May: Marco Furlan (Paris)

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16 May: Balint Virag (Toronto)

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23 May: Patricia Goncalves (Lisbon)

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30 May: Nicolas Perkowski (Humboldt)

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