# Probability Seminar

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

Organisers: Wei Wu, Stefan Adams, Stefan Grosskinsky

###### Term 2 2018-19

**January 9: Benoit Laslier (Paris VII)**

**Title: **Logarithmic variance for uniform homomorphisms on Z^2

**Abstract: **We study random functions from Z^2 to Z that change by exactly 1 between neighboring vertices and show that the variance in the center of a box grows logarithmically with the size of the box, together with various RSW type property for level lines of such function. This model is interesting both as a natural discrete version of taking a continuous function from R^2 to R at random, and also because it is an instance of the 6-vertex model (more precisely the square-ice point) which connects combinatorially (for different values of it's parameters) many well known models such as all FK model, UST or ASEP. The approach does not really on any exact solvability of the model, instead we use a new FKG inequality to adopt the renormalization approach that was developed for the continuity of phase transition for FK. This is joint work with Hugo Duminil-Copin, Matan Harel, Gourab Ray and Aran Raoufi.

**January 16: TALK MOVED TO MS.05**

**Sander Dommers (Hull)
**

**Title: **Metastability in the reversible inclusion process

**Abstract: **In the reversible inclusion process with N particles on a finite graph each particle at a site x jumps to site y at rate (d+η_y)r(x,y), where d is a diffusion parameter, η_y is the number of particles on site y and r(x,y) is the jump rate from x to y of an underlying reversible random walk. When the diffusion d tends to 0 as the number of particles tends to infinity, the particles cluster together to form a condensate. It turns out that these condensates only form on the sites where the underlying random walk spends the most time.

Once such a condensate is formed the particles stick together and the condensate performs a random walk itself on much longer timescales, which can be seen as metastable (or tunnelling) behaviour. We study the rates at which the condensate jumps and show that in the reversible case there are several time scales on which these jumps occur depending on how far (in graph distance) the sites are from each other. This generalises work by Grosskinsky, Redig and Vafayi who study the symmetric case where only one timescale is present. Our analysis is based on the martingale approach by Beltrán and Landim.

This is joint work with Alessandra Bianchi and Cristian Giardinà.

**January 23: Thomas Bothner (King's College)
**

**Title: **When J. Ginibre met E. Schrödinger

**Abstract: **The real Ginibre ensemble consists of square real matrices whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius of a real Ginibe matrix follows a different limiting law for purely real eigenvalues than for non-real ones. Building on previous work by Rider, Sinclair and Poplavskyi, Tribe, Zaboronski, we will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. This system is directly related to several of the most interesting nonlinear evolution equations in 1+1 dimensions which are solvable by the inverse scattering method, for instance the nonlinear Schrödinger equation. The results of this talk are based on the recent preprint arXiv:1808.02419, joint with Jinho Baik.

**January 30: Cyril Labbé (Ceremade)
**

**Title:** Localisation of the continuous Anderson hamiltonian in 1d

**Abstract: **Consider the so-called Anderson hamiltonian obtained by perturbing the Laplacian with a white noise on a segment of size L. This operator is intimately connected to random matrix models and plays an important role for the study of the parabolic Anderson model. I will present a complete description of the bottom of the spectrum of this operator when L goes to infinity. Joint work with Laure Dumaz (Paris Dauphine).

**February 6: Ewain Gwynne (Cambridge)
**

**Title: **The fractal dimension of Liouville quantum gravity: monotonicity, universality, and bounds

**Abstract:** It is an open problem to construct a metric on $\gamma$-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$, except in the special case $\gamma=\sqrt{8/3}$. Nevertheless, the Hausdorff dimension $d_\gamma$ of the conjectural LQG metric is well-defined in the following sense. For a large class of approximations of $\gamma$-LQG distances --- including random planar maps, Liouville first passage percolation, Liouville graph distance, and the Liouville heat kernel --- there is a notion of dimension (in terms of a certain exponent associated with the model) and these exponents all agree with one another.

I will give an overview of some recent progress on understanding $d_\gamma$. In particular, I will discuss the relationships between different exponents, the proof the $\gamma\mapsto d_\gamma$ is strictly increasing, and new upper and lower bounds for $d_\gamma$. These bounds are consistent with (and numerically quite close to) the Watabiki prediction for the value of $d_\gamma$ for $\gamma \in (0,2)$. However, in an extended regime corresponding Liouville first passage percolation with parameter $\xi >2/d_2$, or equivalently LQG with central charge greater than 1, the bounds are inconsistent with the analytic continuation of Watabiki's prediction for certain parameter values.

Based on joint works with Jian Ding, Nina Holden, Tom Hutchcroft, Jason Miller, Josh Pfeffer, and Xin Sun.

**February 13: Nicos Georgiou (Sussex)
**

**Title: **Last passage times in discontinuous environments

**Abstract: **We are studying a last passage percolation model on the two dimensional lattice, where the environment is a field of independent random exponential weights with different parameters. Each variable is associated with a lattice vertex and its parameter is selected according to a discretization of lower semi-continuous parameter function that may admit discontinuities on a set of curves.

We prove a law of large numbers for the sequence of last passage times, defined as the maximum sum of weights which a directed path can collect from (0, 0) to a target point (Nx, Ny) as N tends to infinity and the mesh of the discretisation of the parameter function tends to 0 as 1/N. The LLN is cast in the form of a variational formula, optimised over a given set of macroscopic paths. Properties of maximizers to the variational formula above are investigated in two models where the parameter function allows for analytical tractability. This is joint work with Federico Ciech.

**February 20: George Deligiannidis (Oxford)
**

**Title: **Boundary of the range of a random walk and the Folner property

Abstract: In this work we deal with the question of whether the range of a random walk is almost surely a Folner sequence and show the following results: (a) The size of the inner boundary of the range of recurrent, aperiodic random walks with finite second moment on the two-dimensional integer lattice and of aperiodic, integer-valued random walks in the standard domain of attraction of the symmetric Cauchy distribution, is almost surely of order $n\log^2 (n)$. (b) We establish a formula for the Folner asymptotic of transient co-cycles over an ergodic probability preserving transformation and use it to show that for transient random walk on groups which are not virtually cyclic, for almost every path, the range is not a F ̈olner sequence. (c) For aperiodic random walks in the domain of attraction of symmetric alpha- stable distributions with 1< α≤2, we prove a sharp polynomial upper bound for the decay at infinity of |∂Rn|/|Rn|. This last result shows that the range process of these random walks is almost surely a Folner sequence.

Joint work with Z.Kosloff and S. Gouezel.

**February 27: Pierre-Francois Rodriguez (IHES)
**

**Title:**Sign cluster geometry of the Gaussian free field

**Abstract**: We consider the Gaussian free field on a class of transient weighted graphs G, and show that its sign clusters fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs G belonging to this class include cases in which the random walk on G exhibits anomalous diffusive behavior. Our findings also imply the existence of a nontrivial percolating regime for the vacant set of random interlacements on G. Based on joint work with A. Prévost and A. Drewitz.

**March 6: Jere Koskera (Warwick)
**

**Title**: Asymptotic genealogies of interacting particle systems

**Abstract**: We consider time-evolving, weighted particle systems of fixed size in which a time step consists of a selection stage, during which each particle has a random number of offspring proportional to its weight, and a mutation stage, during which offspring locations are randomly perturbed, and the resulting particles are reweighted based on their new locations. Such interacting particle systems form a rich class of processes with applications in areas including, but not limited to, computational statistics and population genetics. The genealogical tree embedded into the particle system by the selection stages is a key analytical tool, as well as an object of interest in its own right. It is well known that in the neutral case, where particles always have equal weights, the genealogical tree of a fixed number of particles converges to the Kingman coalescent in the infinite system size limit. I will review this classical result, and show that it can be extended to non-neutral models under practically verifiable conditions. This is joint work with Paul Jenkins, Adam Johansen, and Dario Spano.

**March 13: Erik Slivken (Paris)
**

**Title:** Large random pattern-avoiding permutations

**Abstract: **A pattern in a permutation is a subsequence with a specific relative order. What can we say about a typical large random permutation that avoids a particular pattern? We use a variety of approaches. For certain classes we give a geometric description that relates these classes to other types of well-studied concepts like random walks or random trees. Using the right geometric description we can find the the distribution of certain statistics like the number and location of fixed points. This is based on joint work with Christopher Hoffman and Douglas Rizzolo.

###### Term 1 2018-19

**October 3: no seminar
**

**October 10: Milton Jara (IMPA)**

**Title:** Entropy methods in Markov chains

**Abstract:** Building upon Yau's relative entropy method, we derive a new strategy to obtain scaling properties of Markov chains with a large number of components. As an application, we obtain very precise estimates on the mixing properties of a mean-field spin system. Time permitting, we will also discuss the derivation of the speed of convergence of the hydrodynamic limit and non-equilibrium fluctuations of interacting particle systems.

**October 17: Christian Webb (Aalto)**

**Title: **On the statistical behavior of the Riemann zeta function

**Abstract:** A notoriously difficult problem of analytic number theory is to describe the behavior of the Riemann zeta function on the critical line. After reviewing some basic facts about the zeta function, I will discuss what can be said if the problem is relaxed by considering the behavior of the zeta function in the vicinity of a random point on the critical line.

Time permitting, I will also discuss how this problem is related to various models

of probability theory and mathematical physics. The talk is based on joint work with Eero Saksman.

**October 24: no seminar
**

**October 31: Paul Dario (ENS Paris)**

**Title: **Quantitative result on the gradient field model through homogenization.

**Abstract: **Consider the standard uniformly elliptic gradient field model. It was proved by Funaki and Spohn in 1997, by a subadditivity argument, that the finite volume surface tension of this model converges to a limit called the surface tension. The goal of this talk is to show how one can use the tools developed in recent the theory of stochastic homogenization to obtain an algebraic rate of convergence of the surface tension. The analysis relies on the study of dual subadditive quantities, useful in stochastic homogenization, as well as a variational formulation of the partition function and the notion of displacement convexity from the theory of optimal transport.

**November 7: Alexandros Eskenazis (Princeton)**

**Title:** Nonpositive curvature is not coarsely universal

**Abstract: **A complete geodesic metric space of global nonpositive curvature in the sense of Alexandrov is called a Hadamard space. In this talk we will show that there exist metric spaces which do not admit a coarse embedding into any Hadamard space, thus answering a question of Gromov (1993). The main technical contribution of this work lies in the use of metric space valued martingales to derive the metric cotype 2 inequality with sharp scaling parameter for Hadamard spaces. The talk is based on joint work with M. Mendel and A. Naor.

**November 14: David Dereudre (Lille)**

**Title:** DLR equations and rigidity for the Sine-beta process

**Abstract:** We investigate Sine β, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of

one-dimensional log-gases, or β- ensembles, at inverse temperature β > 0. We adopt a statistical physics perspective, and give a description of

Sineβ using the Dobrushin-Landford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine β to a compact set,

conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration.

Moreover, we show that Sineβ is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying

inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough

to include more general long range interactions in arbitrary dimension. (joint work with A. Hardy, M. Maïda and T. Leblé)

**November 21: Nadia Sidorova (UCL)**

**Title: **Localisation and delocalisation in the parabolic Anderson model

**Abstract: **The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It describes the mean-field behaviour of a continuous-time branching random walk. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials it is localised at just one point. However, in a partially symmetric parabolic Anderson model, the one-point localisation breaks down for heavy-tailed potentials and remains unchanged for light-tailed potentials, exhibiting a range of phase transitions.

**November 28: Steffen Dereich (Münster)**

**Abstract: **The concept of quasi-process stems from probabilistic potential theory. Although the notion may not be that familiar nowadays, it is connected to various current developments in probability. For instance, interlacements, as introduced by Sznitman, are Poisson point processes with the intensity measure being a quasi-process. Furthermore, extensions of Markov families as recently derived for certain self-similar Markov processes are closely related to quasi-processes and entrance boundaries.In this talk, we start with a basic introduction of parts of the classical potential theory and then focus on branching Markov chains. The main result will be a spine construction of a branching quasi-process.

The talk is based on joint work with Martin Maiwald (WWU Münster).

**December 5: Weijun Xu (Oxford)
**

**Title:** Weakuniversality of KPZ

**Abstract: **We establish weak universality of the KPZ equation for a class of continuous interface fluctuation models initiated by Hairer-Quastel, but now with general nonlinearities beyond polynomials. Joint work with Martin Hairer.