# 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 3 2020

###### Seminar online with Zoom, Wednesdays at 4:00,

###### Organized by Nikos Zygouras, Leo Rolla together with Sasha Sodin and Queen Mary University of London.

Zoom link for 24/06/2020 : https://www.zoom.us/j/92778010338

**May 6: Boris Khoruzhenko (Queen Mary University of London)**

**Title:** How many stable equilibria will a large complex system have? (slides)

**Abstract:** In 1972 Robert May argued that (generic) complex systems become unstable to small displacements from equilibria as the system complexity increases. May’s model was linear and his outlook was very much local. In search of a global signature of the May instability transition, I will analyse a minimal model for large nonlinear complex systems whereby $N$ degrees of freedom equipped with a stability feedback mechanism are coupled via a smooth homogeneous Gaussian vector field with longitudinal and transverse components. With the increase in complexity (as measured by the number of degrees of freedom and the strength of interaction relative to the relaxation strength), this model undergoes an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial regime of 'absolute instability' where equilibria are on average exponentially abundant, but typically all of them are unstable, unless the interaction is purely longitudinal (purely gradient dynamics). When the complexity increases even further the stable equilibria eventually become on average exponentially abundant unless the interaction is purely transverse (purely solenoidal dynamics). The width of the instability transition region scales as $N^{-1/2}$ (same as in May’s model) and I will argue that in this region the unstable equilibria (saddles) on average have only a very a small proportion of unstable directions which scales as $N^{-1/4}$ with $N$ large.

This talk is based on collaborative works with Yan Fyodorov (PNAS 2016), Gerard Ben Arous and Yan Fyodorov (manuscript in preparation) and Jacek Grela (manuscript in preparation). Our analysis uses Kac-Rice formula for counting zeros of random functions and theory of random matrices applied to the real elliptic ensemble.

**May 13: Vassili Gelfreich (Warwick)**

**Title:** Random models for slow-fast dynamics in absence of ergodicity (slides)

**Abstract:** The equipartition of energy and some other laws of statistical mechanics can be derived from deterministic Hamiltonian equations under the ergodicity assumption. On the other hand, if the Hamiltonian equations possess two different time scales, the ergodic averaging theory can be used to derive adiabatic laws, which are in apparent conflict with the ergodicity of the full system and in particular with the energy equipartition law.

In this talk we discuss some simple deterministic systems which illustrate this phenomenon. We suggest that if the ergodicity of the fast dynamics is violated, the slow dynamics can be modelled with the help of random processes which facilitate transition to the statistical equilibrium for the system.

The talk is based on joint works with V.Rom-Kedar, K.Shah and D.Turaev (partially discussed in the PNAS (2017) paper).

**May 20: Leonardo Rolla (Warwick)**

**Title:** Soliton decomposition of the Box-Ball System (slides)

**Abstract: **The Box-Ball System is a cellular automaton introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg & de Vries (KdV) differential equation. Both systems exhibit solitons, solitary waves that conserve shape and speed even after collision with other solitons. A configuration is a binary function on the integers representing boxes which may contain one ball or be empty. A carrier visits successively boxes from left to right, picking balls from occupied boxes and deposing one ball, if carried, at each visited empty box. Conservation of solitons suggests that this dynamics has many spatially-ergodic invariant measures besides the i.i.d. distribution. Building on Takahashi-Satsuma identification of solitons, we provide a soliton decomposition of the ball configurations and show that the dynamics reduces to a hierarchical translation of the components, finally obtaining an explicit recipe to construct a rich family of invariant measures. We also consider the a.s. asymptotic speed of solitons of each size. An extended version of this abstract, references, simulations, and the slides, all can be found at https://mate.dm.uba.ar/~leorolla/bbs-abstract.pdf . This is a joint work with Pablo A. Ferrari, Chi Nguyen, Minmin Wang.

**May 27: Stephen Muirhead (QMUL)**

**Title: **The phase transition for planar Gaussian percolation models without FKG slides

**Abstract: **Given a smooth stationary centred Gaussian field on the plane and a level , we study the connectivity properties of the set . We prove that the critical level is under only symmetry and (very mild) correlation-decay assumptions, which includes the important example of the random plane wave. Since these models do not satisfy positive associations (the `FKG inequality'), many classical arguments from percolation/statistical mechanics do not apply, and so these are rare example of non-FKG models whose critical point can be rigorously computed. Although many arguments are specific to the Gaussian setting we hope that our techniques may be adapted to analyse other non-FKG models. This is joint work with Hugo Vanneuville and Alejandro Rivera and will appear on arXiv very soon.

**June 3: Vedran Sohinger (Warwick)**

**Title:**Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states (slides)

**Abstract:**Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We are interested in the problem of the derivation of Gibbs measures as mean-field limits of Gibbs states in many-body quantum mechanics.

** **

**June 10: Eleanor Archer**

**Title:**Random walks on decorated Galton-Watson trees slides

**Abstract:**Random trees are the building blocks for a range of probabilistic structures, including percolation clusters on the lattice and a range of statistical physics models on random planar maps. In this talk we consider a random walk on a critical "decorated" Galton-Watson tree, by which we mean that we first sample a critical Galton-Watson tree T, replace each vertex of degree n with an independent copy of a graph G(n), and then glue these inserted graphs along the tree structure of T. We will determine the random walk exponents for this decorated tree model in terms of the exponents for the underlying tree and the exponents for the inserted graphs. We will see that the model undergoes several phase transitions depending on how these exponents balance out.

**June 17: no seminar
**

**June 24: Giuseppe Cannizzaro (Warwick)**

**Title:**2D Anisotropic KPZ at stationarity

**Abstract:**In the context of randomly fluctuating interfaces in (2+1)-dimensions, physicists predict that models possessing three features, a smoothing mechanism, an underlying locally uncorrelated noise and a slope-dependent dynamics, belong to two Universality Classes, the Isotropic and Anosotropic KPZ class whose difference lies in the specifics of the dependence on the slope. We focus on a model belonging to the latter, namely the Anisotropic KPZ (AKPZ) equation. This is a singular stochastic PDE in the critical dimension, which means that, not only it is classically ill-posed but also the recent pathwise techniques for singular SPDEs are not applicable. In the present talk, we consider a regularised version of AKPZ which preserves its invariant measure and we will show that care must be taken when removing the regularisation since more than one scaling regime leads to non-trivial limits.

###### Term 2 2020

**January 8: Marielle Simon (INRIA)**** CANCELLED**

**Title: **Hydrodynamic limit for an activated exclusion process

**January 15: Steffen Dereich (Münster)**

**Title: CLTs for stochastic gradient descent for stable manifolds**

**Abstract: **Nowadays stochastic gradient descent (SGD) algorithms are a standard tool for solving optimization problems. In this talk, we consider the case where the set of local minima is not discrete. This is, for instance, the case in deep learning with ReLU activation function where a single function is parametrized by a non-discrete set of parameters. In this talk we derive central limit theorems for Polyak-Ruppert averaged SGD where the set of local minima form an appropriate stable manifold. We recover the same rate of convergence as in the case of isolated attractors for step-sizes $\gamma_n=n^{-\gamma}$ with $\gamma\in(\frac34,1)$.

**January 22: no seminar**

Statistical aspects of geodesic flows in nonpositive curvature

**January 29: Perla Sousi (Cambridge)**

**Title: Newtonian capacity and forests
**

**Abstract: **A uniform spanning forest of Z^d can be thought of as the ‘’uniform measure’’ on forests (collection of trees) of Z^d. The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest. In joint work with Hutchcroft we calculate the critical exponent for the intrinsic diameter of the past when d=4. Higher dimensions (mean field case) had been calculated previously by Hutchcroft. An important ingredient of the proof is analysing the Newtonian capacity of the range of a loop erased random walk. In this talk I will also survey recent results obtained in collaboration with Asselah and Schapira on the Newtonian capacity of the range of a simple random walk in Z^4.

**February 5: Daniel Ueltschi (Warwick)
**

**Title: Characterising random partitions by random colouring**

**Abstract:** Let be a random partition of the unit interval , i.e. and , and let be i.i.d. Bernoulli random variables of parameter . The *Bernoulli convolution* of the partition is the random variable . The question addressed in this article is: Knowing the distribution of for some fixed , what can we infer about the random partition ? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter is not equal to .

This is joint work with Jakob Björnberg, Cécile Mailler and Peter Mörters.

**February 12: Ofer Busani (Bristol)
**

**Title: **Local stationarity in last passage percolation

**Abstract:** Last passage percolation (LPP) is a family of random growth models in the KPZ universality class, where models are believed to share the same limiting behaviour. One such model is the LPP on the positive quadrant of Z^2; Let the origin be the only infected vertex at time t=0. Place i.i.d. exponentially distributed weights across the vertices of the lattice. The passage time G_x that takes for a vertex x (positioned above and to the right of the origin) to be infected is the maximal weight that can be collected by a path starting from the origin and ending at x and that can only take up-right steps. The unique path that attains the maximum is called geodesic. Now take N large, and consider H_m=G_(N-m,N+m)-G_(N,N) where m << N, i.e. the effect of a small change of the end point on the last passage time. It has been known that H_m has a local Brownian behaviour, in some sense, for large N. We give a strong version of this result, which we refer to as local stationarity. We show how local stationarity can be applied to questions about the limiting behaviour of the KPZ

and the maximal paths.

Joint work with Marton Balazs and Timo Seppalainen.

**February 19: Harald Oberhauser (Oxford)**

**Title:** Learning and testing laws of stochastic processes

**Abstract:** The signature map provides a natural notion of "polynomials on path space" and led to much progress in stochastic analysis. More recently, it has found applications in machine learning for sequence-valued data, such as time-series. I will discuss how this approach can be combined with classic ideas from kernel learning to define a metric for laws of stochastic processes that has several desirable properties. En passant, this requires to answer some questions about learning on non-compact spaces. Further, I will discuss applications in a Bayesian/GP setting and how to deal with the computational complexity with a variational approach.

**February 26: Alex Watson (UCL)**

**Title:** Long-term behaviour of growth-fragmentation processes

**Abstract:** Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly. They are used in models of cell division and protein polymerisation. In the long term, the concentrations of cells with given masses typically increase at some exponential rate and, after compensating for this, they arrive at an asymptotic profile. I will discuss one way to describe this for the average behaviour of the system, pointing out a connection with spectral theory of PDEs. Building on this, I will show that the entire collection of cells, not just their average, also converges to the asymptotic profile.

Joint work with Jean Bertoin.

**March 4: Franco Severo (IHES)**

**Title:**Equality of critical parameters for percolation of Gaussian free field level-sets

**Abstract:**We consider level-sets of the Gaussian free field (GFF) on Z^d, for d ≥ 3, above a given height parameter h ∈ R. As h varies, this defines a canonical site percolation model with slow polynomial decay of correlations. We prove that three natural critical parameters associated to this model, h_∗∗(d), h_∗(d) and \bar{h}(d), respectively describing a strongly non-percolative regime, the emergence of an infinite cluster, and a strongly percolative regime, actually coincide, i.e. h_∗∗(d) = h_∗(d) = \bar{h}(d), for any d ≥ 3. Combined with previous results, this equality has many implications regarding the geometry of GFF level-sets, both in the subcritical and supercritical regimes.

**March 11:** **Marielle Simon (INRIA)**

**Title: **Hydrodynamic limit for an activated exclusion process

**Abstract:** In this talk we will be interested in a one-dimensional exclusion process subject to strong kinetic constraints. More precisely, its stochastic short range interaction exhibits a continuous phase transition to an absorbing state at a critical value of the particle density. We will see that, in the active phase (i.e. for initial profiles smooth enough and uniformly larger than the critical density 1/2), the macroscopic behavior of this microscopic dynamics, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to the class of fast diffusion equations. The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

Based on a joint work with O. Blondel, C. Erignoux and M. Sasada

###### Term 1 2019-20

**October 2: Chiranjib Mukherjee (Münster)**

**Title: **Compactness, Large Deviations and a rigorous theory of the Polaron

**Abstract: **see pdf

**October 9: Benjamin Fehrman (Oxford)**

**Title**: Large deviations in interacting particle systems and stochastic PDE

**Abstract:**In this talk, which is based on joint work with Benjamin Gess, we will draw the link between large deviations in interacting particle systems and large deviations for certain classes of stochastic PDE. The motivating example will be the large deviations of the zero range process about its hydrodynamic limit. We will show informally that the corresponding rate function is identical to the rate function appearing for a degenerate stochastic PDE with nonlinear, conservative noise. Our primary result is a rigorous proof of this fact based upon an intricate treatment of the corresponding skeleton PDE, which is a nonlinear, energy-critical advection-diffusion equation.

**October 16: Hao Shen (Wisconsin)
**

**Title: **Stochastic Ricci flow on surfaces

**Abstract:** We introduce the Stochastic Ricci flow (SRF) in two spatial dimensions. It can be formally written in terms of the evolving Riemannian metric with a space-time noise which is “white” with respect to the metric; or in terms of the conformal factor, so that it becomes a natural quasi-linear generalization of the stochastic heat equation. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus. Modifications are needed for the SRF on general compact surfaces due to conformal anomaly. Under the flow, the total area of the surface follows a squared Bessel process. We also discuss some open questions.

**October 23: John Haslegrave (Warwick)**

**Title**: Phase transition and asymptotics for three-speed ballistic annihilation

**Abstract**: In the ballistic annihilation model, particles are emitted from a Poisson point process on the line, move at constant speed (chosen i.i.d. at initial time) and mutually annihilate when they collide. This model was introduced in the 1990s in physics, but once there are at least three possible speeds little is known rigorously about its behaviour. The most-studied discrete case has speeds of -1, 0 and +1, with symmetric probabilities. Here we prove that a phase transition takes place when stationary particles have probability 1/4, and give precise asymptotics for the decay of particles. This is joint work with Laurent Tournier and the late Vladas Sidoravicius.

**October 30: Tom Hutchcroft (Cambridge)
**

**Title:** Phase transitions in hyperbolic spaces

**Abstract:** Many questions in probability theory concern the way the geometry of a space influences the behaviour of random processes on that space, and in particular how the geometry of a space is affected by random perturbations. One of the simplest models of such a random perturbation is percolation, in which the edges of a graph are either deleted or retained independently at random with retention probability p. We are particularly interested in phase transitions, in which the geometry of the percolated subgraph undergoes a qualitative change as p is varied through some special value. Although percolation has traditionally been studied primarily in the context of Euclidean lattices, the behaviour of percolation in more exotic settings has recently attracted a great deal of attention. In this talk, I will discuss conjectures and results concerning percolation on the Cayley graphs of nonamenable groups and hyperbolic spaces, and give the main ideas behind our recent result that percolation in any transitive hyperbolic graph has a non-trivial phase in which there are infinitely many infinite clusters. The talk is intended to be accessible to a broad audience.

**November 6: David Belius (Basel)**

**Title:**The TAP-Plefka variational principle for mean field spin glasses

**Abstract:**The Thouless-Anderson-Palmer (TAP) approach to the Sherrington-Kirckpatrick mean field spin glass model was described in one of the earliest papers on this model, but has subsequently been a complementary rather than the central component in the theory that has emerged in theoretical physics and mathematics. In this this talk I will recall the TAP approach, and describe how it can be reinterpreted as a variational principle in the spirit of the Gibbs variational principle. Furthermore I will present a rigorous proof of this TAP-Plefka variational principle in the case of the spherical Sherrington-Kirkpatrick model, which allows to compute the free energy based purely on a TAP analysis.

**November 13: Jakob Björnberg (Gothenburg)**

**Title:** The interchange process with reversal

**Abstract:** The interchange process is a model for random permutations formed by composing random transpositions. Here we consider a variant of the interchange process where a fraction of the transpositions are replaced by `reversing transpositions'. A motivation for studying such processes is that they appear in the study of quantum models for magnetism, but they are also interesting in their own right. We will discuss a recent result obtained together with M. Kotowski, B. Lees and P. Milos which describes the scaling limit of the joint distribution of the largest cycles for the process defined on the complete graph.

**November 20: Alexey Bufetov (Bonn)**

**Title:** Color-position symmetry in interacting particle systems

**Abstract:** The asymmetric simple exclusion process (ASEP) is the evolution of a collection of particles on the integer lattice; particles interact according to simple rules and can be of various colors (equivalently, classes). In 2008 Amir-Angel-Valko established an interesting property of such processes: the color-position symmetry. We will discuss a generalization of this result and its new applications to the asymptotic behavior of this class of models.

**November 27: Alisa Knizel (Columbia)**

**Title:**Asymptotics of discrete β-corners processes via discrete loop equations

**Abstract:**We introduce and study stochastic particle ensembles which are natural discretizations of general β-corners processes. We prove that under technical assumptions on a general analytic potential the global fluctuations for the difference between two adjacent levels are asymptotically Gaussian. The covariance is universal and remarkably differs from its counterpart in random matrix theory. Our main tools are certain novel algebraic identities that we introduce. Based on joint work with Evgeni Dimitrov (Columbia University)

**December 4: No Seminar**