# 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: Quentin Berger (Paris 6)

**Title:** Directed polymers in random heavy-tail environment

**Abstract:** The directed polymer model, introduced more than 30 years ago and intensively studied since then, can be used to describe a polymer interacting with the impurities of a heterogeneous medium. I will present a brief history of the model, and I will focus on the localization phenomenon of the trajectories, the polymer ‘’stretching’’ to reach more favorable regions of the environment. Describing the localized trajectories (super-diffusivity exponent, scaling limits, etc…) is mostly open, but I will consider the case of an environment with heavy-tail distribution, where these results are at reach.

(Joint work with Niccolò Torri)

14 March: Marcin Lis (Cambridge)

**Title**: The double random current nesting field

**Abstract:** A configuration of the planar random current model can be viewed as a

collection of dual Ising contours together with an independent Bernoulli

bond percolation with prescribed success probabilities. The double

random current model is simply a superimposition of two iid random

current configurations. Its clusters are composed of XOR-Ising contours

and of additional components arising from the percolation process or two

overlapping single Ising contours. For each such cluster C we toss an

independent +-1 symmetric coin X_C. A cluster C is called odd around a

face u if the contours contained in C assign spin -1 to u under +1

boundary conditions. The double random current nesting field at u is

defined to be the sum of X_C over clusters C odd around u.

I will provide a measure-preserving map between double currents and

dimers on a particular bipartite graph. Under this map the nesting field

becomes the height function of the dimer model. Using this connection

together with the results of Kenyon, Okounkov and Sheffield on the dimer

model, I will prove that the magnetization of the critical Ising model

on any biperiodic graph vanishes.

This is joint work with Hugo Duminil-Copin.

**Term 3, 2017-2018**

25 April: Jan Maas (IST, Vienna)

**Title**: Gradient flows and quantum entropy inequalities via matrix optimal transport

**Abstract**: We present a new class of transport metrics for density matrices, which can be viewed as non-commutative analogues of the 2-Wasserstein metric. With respect to these metrics, we show that dissipative quantum systems can be formulated as gradient flows for the von Neumann entropy under a detailed balance assumption. We also present geodesic convexity results for the von Neumann entropy in several interesting situations. These results rely on an intertwining approach for the semigroup combined with suitable matrix trace inequalities. This is joint work with Eric Carlen.

2 May: Yvan Velenik (Geneva)

**Title:** Two applications of the Ornstein-Zernike theory to Ising and Potts models

**Abstract:** I'll discuss two problems related to the decay of correlations in d-dimensional Ising and Potts models above their critical temperature. In the first, we consider the effect of a defect line on the correlation length of the model. In the second, we analyze, for the Ising model, the prefactor to the exponential decay for covariances of general local observables. The Ornstein-Zernike theory lies at the core of our approach to both problems.(This is based on joint works with Sébastien Ott.)

9 May - extra seminar: Peter Friz (TU Berlin & WIAS)

**Title:** Varieties, Signature Tensors, Rough Paths

**Abstract:** The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures. Joint work with Carlos Améndola and Bernd Sturmfels

9 May: Marco Furlan (Paris)

**Title:** Weak universality for a class of 3d stochastic reaction-diffusion models

**Abstract:** Recently M. Hairer and W. Xu (ArXiv: 1601.05138) obtained a convergence result for a class of reaction-diffusion models with stochastic forcing term. Using regularity structures they showed that for a Gaussian forcing term converging to the white noise, and a small (compared to the noise) reaction term which is an odd polynomial, the family converges to the $\Phi^4_3$ model.

In my talk I will show how to extend this result to more general reaction terms (with only a finite number of derivatives and exponential growth), using Malliavin calculus to bound enhanced noise terms in a paracontrolled distributions setting. The result was obtained in a joint work with M. Gubinelli (ArXiv: 1708.03118).

16 May: Andre Schlichting (Aachen)

**Title:** Phase transitions for the McKean-Vlasov equation on the torus.

**Abstract:** In the talk, the McKean-Vlasov equation on the flat torus is studied. The model is obtained as the mean field limit of a system of interacting diffusion processes enclosed in a periodic box. The system acts as a model for several real-world phenomena from statistical physics, opinion dynamics, collective behaviour, and stellar dynamics. This work provides a systematic approach to the qualitative and quantitative analysis of the McKean-Vlasov equation. We comment on the longtime behaviour and convergence to equilibrium, for which we introduce a notion of H-stability. The main part of the talk considers the stationary problem. We show that the system exhibits multiple equilibria which arise from the uniform state through continuous bifurcations, under certain assumptions on the interaction potential. Finally, criteria for the classification of continuous and discontinuous transitions of this system are provided. This classification is based on a fine analysis of the free energy. The results are illustrated by proving and extending results for a wide range of models, including the noisy Kuramoto model, Hegselmann-Krause model, and Keller-Segel model. (joint work with José Carrillo, Rishabh Gvalani, and Greg Pavliotis)

23 May: Julian Fischer (IST Austria)

**Title:** The choice of representative volumes for random materials

**Abstract:** The most widely employed method for determining the effective large-scale properties of random materials is the representative volume element (RVE) method: It basically proceeds by choosing a sample of the random medium - the representative volume element - and computing its properties. To obtain an accurate approximation for the effective material properties, the RVE should reflect the statistical properties of the material well. Hence, it is desirable to choose a large sample of the random medium as an RVE. However, an increased size of the RVE comes with an increased computation cost. For this reason, there have been attempts in material science and mechanics towards capturing the statistical properties of the material in a better way in an RVE of a fixed size. We provide an analysis of an approach by Le Bris, Legoll, and Minvielle, which has been capable of improving the computational efficiency by a factor of 10-50 in some numerical examples by such an ansatz.

30 May: Nicolas Perkowski (Humboldt)

**Title:** Additive functionals as rough paths

**Abstract**: I will present an invariance principle for additive functionals of Markov processes in rough path topology. The limit of the iterated integrals coincides with the Statonovich iterated integrals of the limiting Brownian motion, plus a correction term that only vanishes if our Markov process is reversible. As an application we derive the rough path scaling limit of a random walk with random conductances. This is joint work with Jean-Dominique Deuschel and Tal Orenshtein.

6 June: Minmin Wang (Bath)

**Title:** The Box-Ball System, some combinatorial representations, and invariant measures

**Abstract:** The Box-Ball system was first introduced by two physicists Takahashi and Satsuma as a discrete system which exhibits a phenomenon of solitons (substructures which conserve their shapes and speeds even after collisions with the other solitons).

Its version in Z runs as follows. The configuration space is {0, 1}^Z, representing boxes indexed by Z which may contain one ball or be empty. A carrier visits successively these boxes from left to right, picking up balls and then putting them down into the first available boxes.

We will discuss some combinatorial representations of the system and a connection with Pitman’s 2M-X theorem. We will also talk about a decomposition of the system into soliton components and how from this decomposition we can build a rather rich family of invariant measures.

This is based on a joint work with Pablo Ferrari, Chi Nguyen and Leo Rolla.

20 June: Michail Loulakis (Athens)

**Title:** The stochastic shallow lake problem

**Abstract:** We study the welfare function and the optimal control of the phosphorus deposition for the shallow lake problem with multiplicative noise. We show that the welfare function is the viscosity solution of the associated Bellman equation and we establish several properties including its asymptotic behaviour at infinity. Finally we examine the trajectories of the optimally controlled lake.

Joint work with George Kossioris (Crete) and Panagiotis Souganidis (Chicago)

27June: Xue-Mei Li (Imperial)

**Title: **Yet another look at Stochastic Averaging

**Abstract: **

Stochastic averaging goes back to the 60’s. It keeps on reinventing itself and find its use in unexpected places. I’ll report some of the recent progress.