Exclusion processes: some recent results and open questions
We define exclusion processes on lattices and also more generally on (possibly random) graphs. We give some results on invariant measures and on the speed of the tagged particle. Turning to (sequences of) finite graphs, we study mixing times to describe how fast the processes approach their invariant laws. In order to do so, we discuss different current regimes and the behaviour of second class particles. The minicourse has a focus on open problems and welcomes questions coming from the participants.
Disordered spin systems, first-passage percolation and minimal surfaces in random environment
The course will explore various ways in which disorder (i.e., a random environment) may alter the behavior of familiar models. Emphasis will be put on the many open questions in the field, which the audience is encouraged to contemplate. We first discuss disordered spin systems, with a focus on the random-field Ising model, explaining cases in which the disorder destroys the long-range order of the model and cases in which it does not. We then move to discuss the random geometry obtained by perturbing Euclidean, or lattice, geometry with random, independent, local perturbations. We will first consider first-passage percolation, which studies the geodesics of this random geometry, and finally discuss minimal surfaces in the random geometry. No prior knowledge in these topics will be assumed.
Probability on hierarchical lattices
In this mini-course we will study stochastic processes on a special class of graphs called hierarchical lattices. The special symmetries enjoyed by these structures allow us to perform explicit calculations using a variety of tools, specially coming from dynamical systems. For one thing, this can serve as an inspiration to investigate renormalization groups. Moreover, the surprising precision that one is able to attain using such techniques, makes hierarchical lattices an excellent test-bed for making predictions about various models elsewhere.
Non-commutative singular SPDE
In this talk, I will describe some recent progress on singular stochastic partial differential equations in the setting of non-commutative probability theory - examples will include the stochastic quantization of Fermionic quantum field theories and also the setting of free probability. This is based on joint work with Martin Hairer and Martin Peev.
Uniqueness of the invariant measure of the phi42 dynamics in infinite volume
We consider the phi42 dynamics in infinite volume, i.e. on the full plane, and study its invariant measures. In finite volume those are known to be unique and given by the phi42 field theory. In infinite volume uniqueness may fail, for instance due to phase transitions. We prove uniqueness whenever the susceptibility of the finite volume phi42 measure remains bounded, a presumably optimal criterion. This is done by adapting to the field-theory setting the Holley-Stroock-Zegarlinski approach to uniqueness for statistical mechanics models. This approach is based on a volume-independent bound on the log-Sobolev constant of the associated dynamics, together with crude, model-independent bounds. In the phi42 case the log-Sobolev has recently been established, but substantial work is required to adapt the other parts of the argument. The talk is based on joint work in progress with R. Bauerschmidt and H. Weber.
Weaves, webs and flows
We consider "weaves" - loosely, a weave is a set of non-crossing cadlag paths that covers 1+1 dimensional space-time. Here, we do not require any particular distribution for the particle motions. Weaves are a general class of random processes, of which the Brownian web is a canonical example; just as Brownian motion is a canonical example of a (single) random path. It turns out that the space of weaves has an interesting geometric structure in its own right, which will be the focus of the talk. This structure provides key information that leads to a weak convergence theory for general weaves. Joint work with Jan Swart, based on https://doi.org/10.1214/24-EJP1161.
Stochastic and geodesic completeness for landmark space
In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by n ≥ 2 distinct landmark points in R^d. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimisation problem which minimises a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold Q of n distinct landmark points in R^d can be endowed with a Riemannian metric g such that the above optimisation problem is equivalent to the geodesic boundary value problem for g on Q. Despite its importance for modelling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold (Q,g) is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterisation of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel. I will further discuss joint work with Stephen C. Preston and Stefan Sommer which, for any number of landmarks in R^d and again with respect to a radial kernel, provides a sharp criterion guaranteeing geodesic completeness or geodesic incompleteness, respectively, of (Q,g).
Conditioned exit measures for branching Markov processes
Motivated by the need to understand rare events for fissile systems, we consider path decompositions for conditioned exit measures of a general class of branching Markov processes. In particular, we show that conditioning these processes to exit a domain via certain sets leads to a many-to-few decomposition. This is joint work with Chris Dean, Simon Harris and Andreas Kyprianou.
A localisation phase transition for the catalytic branching random walk.
In this joint work with Bruno Schapira, we show the existence of a phase transition between a localisation and a non-localisation regime for a branching random walk with a catalyst at the origin. More precisely, we consider a continuous-time branching random walk that jumps at rate one, with simple random walk jumps on ℤd, and that branches (with binary branching) at rate λ>0 everywhere, except at the origin, where it branches at rate λ0>λ. We show that, if λ0 is large enough, then the occupation measure of the branching random walk localises (i.e. converges almost surely without spatial renormalisation), whereas, if λ0 is close enough to λ, then localisation cannot occur, at least not in a strong sense.
Stake-governed random tug-of-war and the infinity-Laplacian
Imagine a version of chess, where each player has 100 euros, and, instead of alternating moves, before each turn they stake some portion of their fortunes, then flip a coin that is biased according to the stakes, and the winner of the coin toss makes the next move. A metaphor for how to convert economic position into positional advantage. This would of course be too difficult to analyze mathematically. Instead, consider random tug-of-war on graphs, which is a probabilistic game that was introduced by Peres-Schramm-Sheffield-Wilson (JAMS 2009) to aid the analysis of the infinity-Laplace equation, a singular elliptic PDE. We introduce the stake-governed version of this game, and solve it on finite rooted trees, by finding the Nash equilibria for the stakes and moves. Joint work with Yujie Fu and Alan Hammond.
Branching Internal DLA
Internal DLA is a random aggregation process in which the growth of discrete clusters is governed by the harmonic measure seen from an internal point. That is, a simple random walk is released from inside the cluster, and its exit location is added to it. The asymptotic shape of IDLA on Zd starting from a single seed has long been known to be a Euclidean ball, with very small fluctuations. In this talk I will discuss a natural variant of IDLA, namely Branching IDLA, in which the particles that drive the process perform critical branching random walks rather than simple random walks. We will show that BIDLA has a strikingly different phenomenology, namely we prove a phase transition from macroscopic fluctuations in low dimension to the existence of a shape theorem in higher dimension. Based on a joint work with Amine Asselah (Paris-Est Créteil) and Lorenzo Taggi (Rome La Sapienza).
Spanning Trees in the Euclidean Random Assignment Problem
We consider the Optimal Transport (in the sense of Monge and Kantorovich) among two independent sets of i.i.d. points on a domain of the plane. We briefly review how non-rigorous methods of Field Theory led to conjecture the leading term in the average cost, and a relationbetween the optimal transportation map and the solution of the Poisson's Equation (with Neumann b.c.) for the difference of the twomeasures. We describe two combinatorial procedures that allow to construct spanning trees naturally associated to this problem. In the first case we have a tree with "free boundary conditions", while in the second case we have "wired boundary conditions". We show how the field-theoretical approach allows to predict certain statistics on these trees. We observe that these predictions are in accord with the conjecture that the two trees are asymptotically distributed as the Uniform Spanning Tree, with corresponding boundary conditions. Based on work in progress with Sergio Caracciolo and Gabriele Sicuro.
Non-monotone phase transition in interacting particle systems
In this talk we will discuss a reaction-diffusion particle system which has a non-monotone phase transition. I will explain the techniques used to analyze monotone models and how they can be refined to analyze non-monotone particle systems. Based on joint works with Leandro Chiarini and Tom Finn.
Random walk above a concave obstacle: curvature effects
Let f:[-1,1]→R be uniformly strictly concave and C3. I'll consider a rather general one-dimensional random walk
(Sn; -N≤n≤N) conditioned on S-N=⌈Nf(-1)⌉, SN=⌈Nf(1)⌉ and for all -N≤n≤N, Sn≥Nf(n/N).
I'll explain how one can show that the typical height of the random walk above this obstacle, Sn- Nf(n/N) is of order N1/3. I'll then discuss what happens when one relaxes the assumptions on f. Namely, in the special case f(x)=-|x|ɑ with α in [1,∞), I'll show that the typical height above 0 is of order N(ɑ-1)/(2ɑ-1). I'll end the talk explaining our original motivation for this problem, originating from equilibrium statistical mechanics: analyzing the behavior of the interface of the planar Ising model along a macroscopically curved boundary. This is a joint work with Sébastien Ott.
Phase Transition in the Monkey Walk
The Monkey Walk is a random walk model introduced by Boyer and Solis-Salas ('14) to model the behaviour of Capuchin monkeys. In particular it introduces non-Markovian relocation events which require knowledge of the entire history of the walk. Mailler and Bravo ('19) showed that the occupation measure of the Monkey walk exhibits slow diffusivity under a mild set of ergodicity conditions. However, Boyer et al ('19) stated that, by slightly altering the likelihood of relocation at a single site, we introduce a phase transition between diffusivity and localisation. We are able to prove via Measure-Valued Polya Processes that the occupation measure does indeed localise. (This is joint work Daniel Kious and Cecile Mailler).
Random cluster model on trees and tree recursions
The study of certain statistical mechanics models on trees can sometimes be reduced to the study of a "simple" recursion, as is the case for the random cluster model. It turns out that when this recursion is concave, it can be compared to that of effective conductances (potentially nonlinear) between the root and the leaves of the tree. In collaboration with Quentin Berger (LAGA), we estimated the precise asymptotic behavior of nonlinear conductances on Galton-Watson trees, which allowed us to obtain detailed information about the random cluster model on random trees.
Poisson processes of lines/roads and random geometry
We consider the "driving time" random metric induced by a Poisson process of roads (i.e, lines with speed limits) in Euclidean space. This model was introduced by Aldous and Kendall a few years ago. As the Brownian sphere or other random metric spaces, it exhibits nice fractal properties: almost surely, the metric space induced by the Poisson process of roads is homeomorphic to the underlying Euclidean space, and its Hausdorff dimension can be calculated to be a constant strictly greater than the "topological" dimension. Another interesting feature of this random metric is the properties of its geodesics: (existence,) uniqueness between typical points, confluence... In the planar case, we proved recently (with Nicolas Curien and Jonas Kahn) that geodesics do not "pause en route", i.e, do not use roads with arbitrarily small speed except at their endpoints, as conjectured by Kendall. Building on this, we study the geodesic frame, stars and "hubs", and give a complete description of the local structure of geodesics around points on roads.
Simulating Random Systems Without Simulating Random Systems
Many random systems are large. Many are complex. Plenty are both. This can present significant difficulty when attempting to obtain numerical data as experiments can take days, weeks, or even months for sufficiently large systems. In some cases, direct simulation can be completely intractable. In this talk, I will present an algorithm - Linear Logarithmic Relaxation - which allows us to extract unbiased numerical estimates for the expectation values of system observables for multiple parameter values and observables with just a single batch of computational effort. This is done via estimation of the density of energy states within the system using an adapted version of the Robbins-Monro algorithm and importance sampling. This talk is based on work with Professor Amanda Turner (Leeds) and Professor Kurt Langfeld (Western Sydney).
Multidimensional Reflected McKean-Vlasov Stochastic Differential Equations
In this talk, we investigate the multidimensional Skorokhod problem with normal reflection in a family of time-dependent convex domains that are càdlàg with respect to the Hausdorff metric. We then show the existence and uniqueness of solutions to multidimensional McKean-Vlasov stochastic differential equations reflected in these time-dependent domains. Additionally, we derive stability properties with respect to the initial condition and the coefficients. Finally, we establish a propagation of chaos result.
Limiting behaviour of geometric functionals of convex hulls spanned by planar random walks
In this talk, we study the limiting behavior of the perimeter and diameter functionals of the convex hull spanned by the first n steps of two planar random walks. As the main results, we obtain the strong law of large numbers and the central limit theorem for the perimeter and diameter of these random sets.