Colloquium in probability

The rotor-router model is a deterministic process, in which we place an arrow at every vertex of the underlying graph G, which points to one of its neighbours. A particle then moves on our graph, by first turning the rotor at its current location based on a deterministc ruleset, and then moving towards the new direction of the rotor. A natural generalization of this model is then given, by allowing the turn of the rotor to be itself a random outcome, depending only on the current direction of the rotor. This leads us to defining locally Markov walks, which are stochastic processes, whose next step only depends on the last action the particle performed at its current location. In my talk we will thoroughly define the rotor-router model and discuss one of the main conjectures concerning the behaviour of the walkers, which is whether the rotor-router model with initial directions of the rotors chosen uniformly at random is recurrent on the two-dimensional integer grid. We will also introduce locally Markov walks, and discuss some results of locally Markov walks on finite graphs, as well as several open problems to consider for future research.

In a recent paper in the journal Electronic Communications in Probability, Lanchier and Mercer discussed a variant of the original Deffuant model that additionally featured repulsion for individuals on the integer line, holding opinions farther than a confidence threshold \(\theta\) apart. The authors proved that all non-trivial choices of the parameter \(\theta\) resulted in the divergence of the opinion gap along at least one edge, meaning consensus never arises in this model. Inspired by a phenomenon where individuals of too different opinions may stop communicating, we introduce a new model with an added parameter \(K>\theta\), where repulsion stops after the opinion gap on an edge exceeds \(K\), 'deactivating' the edge so no interaction occurs. On this model still no global consensus occurs, however both agreeing opinion clusters and 'inactive' clusters arise.

There have been a lot of recent progress on branching particle systems with selection, in particular on the N-particle branching random walk (N-BRW). In the N-BRW, N particles have locations on the real line at all times. At each time step, every particle generates a number of children, and each child has a random displacement from its parent's location. Then among the children only the N rightmost are selected to survive and reproduce in the next generation. In this talk we will investigate a noisy version of the N-BRW. In this model the N surviving particles are selected at random from the children in such a way, that particles more to the right on the real line are more likely to be selected. I will present some recent results on the asymptotic behaviour of this particle system as N goes to infinity; including the distribution of the N particles on the real line and the speed of the particle cloud. Our results show that as we change the selection parameter, there is an interesting phase transition in these asymptotic properties. This is joint work with Colin Desmarais, Bastien Mallein, Francesco Paparella and Emmanuel Schertzer.

In this talk, we present two novel variants of the contact process. In the first variant individuals carry a viral load. An individual with viral load zero is classified as healthy and otherwise infected. If an individual becomes infected it begins with a viral load of one, which then evolves according to a Birth-Death process. In this model, viral load indicates severity of the infection such that individuals with a higher load can be more infectious. Moreover, the recovery times of individual is not necessarily exponentially distributed and can even be chosen to follow a power-law distribution. In the second variant individuals are permanently infected albeit in two states: actively infected or dormant. The dynamics of these individual states are again governed by a Birth-Death process. Dormant infections do not interact with neighbouring individuals but may reactivate spontaneously. Active infections reactivate dormant neighbours at a constant rate and may become dormant themselves. We present a Poisson construction for both variants. For the first model, we study the phase transition of survival and discuss existence of a non-trivial upper invariant law. Additionally, we derive a duality relationship between the two variant, which we use to uncover a phase transition regarding invariant distributions in the second variant.

In this talk, I will present selected developments from the past decade on the geometry of random polytopes. Particular emphasis will be placed on fluctuation results, both those obtained by means of Stein’s method and those derived through alternative approaches. I will also highlight recent progress in the planar setting, where techniques from analytic combinatorics have opened up new perspectives.

We introduce a model of random walks on Z^3 with random orientations of lines. This model can be seen as a 3D-version of a model of diffusion in inhomogeneous porous medium that has been introduced by Matheron and de Marsily. This 3D-model is related to the new process of iterated random walk in random sceneries (PAPAPA in french). We establish a joint limit theorem for the random walk (PA in french), the random walk in random sceneries (PAPA in french), and the iterated random walk (PAPAPA). This result is a joint work with Nadine Guillotin-Plantard and Frédérique Watbled. We will explain the relation between this work and previous developments for random walks in random sceneries. We will also present a conjecture about iterated random walks of higher order, and discuss about the difficulties to establish this conjecture.

The Cluster-cluster model was defined by Meakin in 1984. Consider a stochastic process on the graph Z^d. Each x in Z^d starts with a cluster of size 1 with probability p in (0,1] independently. Each cluster C performs a continuous time SRW with rate |C|^{-\alpha}. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. In this talk we will present results about explosion, non-explosion and cluster growth rates, as a function of the dimension - d, percolation density - p and diffusion rate parameter - alpha. Joint work with Noam Berger (TUM), Eviatar B. Procaccia (Technion) and Dominik Schmid (Augsburg University).

Consider a Brownian motion $B=(B_t)_{t \ge 0}$ as well as a positive random variable $\xi$ independent of $B$ and a measurable, locally bounded function $u: \R \times [0,\infty) \to [0,\infty)$. Let $$\tau:= \inf\left\{T \ge 0: \int_0^T u(B_s,s) \D s \ge \xi\right\}$$ be the first time the time-inhomogeneous additive Brownian functional associated with $u$ reaches the threshold $\xi$. We will analyze the asymptotic behavior of $\p(\tau \gne T)$ as $T \to \infty$ and, in particular, provide sufficient criteria for this probability to decay like a multiple of $\frac{1}{\sqrt{T}}$. Subsequently, we will discuss the existence and long-term behavior of the associated conditioned process, i.e., of $B$ conditioned on the rare event $$\{\tau=\infty\} = \left\{\int_0^t u(B_s,s) \D s \lne \xi \text{ for all } t \ge 0\right\}.$$ Our framework, in particular, covers occupation times below any moving barrier dominated in modulus by $t^\gamma$ for some $\gamma \lne \frac{1}{2}$ as $t \to \infty$. Further, it covers the case where $u$ is a modified solution of the FKPP equation. This will be the key to upcoming results concerning branching Brownian motions with critically large maximum, a joint project with Bastien Mallein (Toulouse).

After motivating the Stefan problem from the random growth model perspective, I will discuss its discontinuities in time. These turn out to be characterized by the cascade equation, a second-order hyperbolic PDE. Questions of existence and regularity for the latter can be answered by expressing its solution as the value function of a player in an equilibrium of a suitable mean field game. Based on joint work with Yucheng Guo and Sergey Nadtochiy.

Multispecies asymmetric exclusion processes (ASEPs) are interacting particle systems characterised by simple, local dynamics, where particles occupy lattice sites and interact only with their adjacent neighbors, following asymmetric exchange rules based on their species labels. I will present recent results on two-point correlation functions in multispecies ASEPs, including models on finite rings and their continuous-space limit as the number of sites tends to infinity. Using combinatorial tools such as Ferrari–Martin multiline queues, projection techniques, and bijective arguments, we derive exact formulas for adjacent particle correlations and resolve a conjecture in the continuous multispecies TASEP (Aas and Linusson, AIHPD 2018). We also extend finite-ring results of Ayyer and Linusson (Trans AMS, 2017) to the partially asymmetric case (PASEP), formulating new correlation functions that depend on the asymmetry parameter. I will briefly outline ongoing work on boundary-driven multispecies B-TASEP and long-time limiting states in periodic ASEPs, suggesting connections between pairwise correlations and stationary-state structure.

Self-interacting Markov chains arise in a range of models and applications. For example, they can be used to approximate the quasi-stationary distributions of irreducible Markov chains and to model random walks with edge or vertex reinforcement. The term self-interacting Markov chain is something of a misnomer, as such processes interact with their full path history at each time instant, and therefore are non-Markovian. Under conditions on the self-interaction mechanism, we establish a large deviation principle for the empirical measure of self-interacting chains on finite spaces. In this setting, the rate function takes a strikingly different form than the classical Donsker-Varadhan rate function associated with the empirical measure of a Markov chain; the rate function for self-interacting chains is typically non-convex and is given through a dynamical variational formula with an infinite horizon discounted objective function.

A central question in Markov chain mixing is the occurrence of cutoff, a phenomenon according to which a Markov chain converges abruptly to the stationary measure. The focus of this talk is the limit profile of a Markov chain that exhibits cutoff, which captures the exact shape of the distance of the Markov chain from stationarity. We will discuss techniques for determining the limit profile and its continuity properties under appropriate conditions.

The Vertex Reinforced Jump Process (VRJP) is a self-reinforcing stochastic process on a graph. There are open questions about its behaviour on Z^d, for example, whether there exists a unique phase transition for d > 2. In order to help build an intuition for these questions, we develop an efficient algorithm for the simulation of the VRJP's random environment on large grid graphs, which can serve as approximations to its dynamics on Z^d. By exploiting the properties of the VRJP's beta-field, we propose an iterative algorithm for its efficient simulation. Its complexity is linear in the number of vertices and cubic in the maximal vertex degree. The random environment can be derived from this beta field by solving a linear system, whose size equals the number of vertices in the graph. To solve this linear system, we use Conjugate Gradients and give a brief discussion of some preconditioning and optimization strategies based on the adjacency matrices of the grid graphs

We revisit the classic simple symmetric exclusion process, which has the same hydrodynamic limit as a system of independent random walkers. Our aim is to provide a deeper understanding why the exclusion mechanism does not influence the hydrodynamic limit. We do this by interpreting each time the exclusion mechanism is invoked as a collision between particles, then keep track of the number of collisions in the system and pass to the hydrodynamic limit. In fact we study four of such variables under different scaling regimes and obtain a zoo of hydrodynamic limits - some deterministic and some stochastic.