Colloquium in probability

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.

The frog model is a classical branching model for the spread of an infection. In this model, sleeping frogs are placed on the vertices of a graph and initially one vertex is activated. Active frogs move as simple random walks and wake up all sleeping frogs they encounter. Motivated by the addition of an immunological response, we present an extension of this model in which sleeping frogs must be visited a random number of times, i.i.d. as some $I$, before they awaken, and the active frogs that attempt to wake them up are killed. We examine the propagation speed and demonstrate that the frogs spread ballistically for a $2+\varepsilon$-moment assumption on $I$, but sublinearly for a more heavy-tailed $I$. By constructing a series of renewal times, at which the front becomes independent of the past, we are able to derive a shape theorem under more technical assumptions.

Random fields of gradients on the lattice are a class of model systems arising in the study of random interfaces, random geometry, field theory and elasticity theory. The gradient fields we consider are characterised by an imposed boundary tilt and the free energy (called surface tension in the context of random interface models) as a function of tilt. Two questions of interest are whether the surface tension is strictly convex and whether the large-scale behaviour of the model is that of the massless free field (Gaussian universality class). Where the Hamiltonian of the system is determined by a strictly convex potential, good progress has been made on these questions over the last three decades. For models with non-convex energy fewer results are known. Open problems include the conjecture (verified recently in the strictly convex case) that, in any regime where the scaling limit is Gaussian, its covariance (diffusion) matrix should be given by the Hessian of surface tension as a function of tilt. I will survey some recent advances in this direction using renormalisation group arguments and describe our result confirming the conjectured behaviour of the scaling limit on the torus and in infinite volume for a class of non-convex potentials in the regime of low temperatures and small tilt. This is based on joint work with Stefan Adams.

The abelian sandpile model introduced by Per Bak, Chao Tang and Kurt Wiesenfeld was the first discovered example of a dynamical system exhibiting self-organized criticality. Since its introduction in 1987, the model has seen widespread research interest from mathematicians and physicists alike, with a focus on explaining the complex global behaviour that emerges from the interplay of the local toppling rules. In my talk, I will introduce the abelian sandpile model, its toppling rules and how we use these dynamics to define the abelian sandpile Markov chain. We will cover the most important aspects of the sandpile Markov chain, and discuss how these apply to modern research questions on the abelian sandpile model about the distribution of particles and avalanche sizes, with a focus on how the model behaves on fractal state spaces.

We consider a stochastic process on the graph $\mathds{Z}^d$. Each $x\in \mathds{Z}^d$ starts with a cluster of size 1 with probability $p \in (0,1]$ independently. Each cluster $C$ of performs a continuous time SRW with rate $\abs{C}^{-\alpha}$. 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. Focusing on dimension $d=1$, we show that for $\alpha>-2$, at time $t$, the cluster size is of order $t^\frac{1}{\alpha + 2}$, and for $\alpha \le -2$ we get an infinite component. Additionally, for $\alpha = 0$ we show convergence in distribution of the scaling limit.

A sum rule is an identity connecting the entropy of a measure with the coefficients involved in the construction of its orthogonal polynomials. It is possible to prove sum rules using large deviation theory. We consider the weighted spectral measure of random matrices and prove a large deviation principle when the size of the matrix tends to infinity. The measure may be described by its spectral information or its recursion coefficients. This allows to write the rate function in two different ways, which leads to the sum rule. In this talk I present an extension to unitary random matrices in the multi-cut case, when the limit of the spectral measure is supported by several arcs of the unit circle. In this case the rate function cannot be given explicitly. We can still state a sum rule under additional conditions on the recursion coefficients, which are related to finding a specific representative in the Aleksandrov class of measures. The talk is based on a joint work with Fabrice Gamboa and Alain Rouault.

In this presentation, we will study a covering process in the discrete one-dimensional torus that uses connected arcs of random sizes. More precisely, we fix a distribution $\mu$ on $\mathbb{N}$ and for every $n\geq 1$ we will cover the torus $\mathbb{Z}/n\mathbb{Z}$ as follows: at each time step, we place an arc with a length distributed as $\mu$ and in a uniform starting point. Eventually, the space will be entirely covered by these arcs. Changing the arc length distribution $\mu$ can potentially change the limiting behavior of the covering time. In this lecture, we will expose four distinct phases for the fluctuations of the cover time in the limit. These phases can be informally described as the Gumbel phase, the compactly support phase, the pre-exponential phase, and the exponential phase. Furthermore, we expose a continuous-time cover process that works as a limiting distribution within the compactly support phase.

The Ising Model is a classical model in statistical physics describing the behavior of ferromagnetic moments on a lattice interacting via a local interaction. When the lattice is one-dimensional and in the case of homogeneous nearest-neighbor interaction, the model is known to be exactly solvable (and simple). However, the disordered version of the one-dimensional Ising Model (called the Random Field Ising Chain), where the chain interacts with an i.i.d environment, is a much more challenging model. In particular, it exhibits a pseudo-phase transition as the strength Gamma of the inner-interaction goes to infinity. A description of the typical configurations when Gamma is large has been given in the physical literature in terms of a renormalisation group fixed point. In this talk, we will present and discuss the RFIC model, on the level of the free energy and on the level of configurations. We will consider the cases of both centered and uncentered external fields. The notion of Gamma-extrema of the Brownian motion, introduced by Neveu and Pitman, will play a crucial role in our analysis.

Pattern formation phenomena are omnipresent in the natural sciences (especially chemistry, biology, and physics). Consequently, patterns, their formation, and their stability have been studied extensively in the theory of dynamical systems. However, while the deterministic side is in large parts well understood, very little is known on the stochastic side. We want to use geometric methods and techniques from dynamical systems and regularity structures to extend certain deterministic stability results to generalised patterns in singular SPDEs.

No-Free-Lunch theorems are important results in the mathematical foundations of statistical learning. They typically state that, in expectation w.r.t. a uniformly chosen target concept, no machine learning algorithm performs better on unseen data than random guessing. Put differently, one algorithm can only outperform another when being supplied with sufficient a priori knowledge by means of training data or design. In this talk, I will present a new kind of No-Free-Lunch theorem, namely for so-called autoregressive models, most prominently used in Large Language models powering, e.g., OpenAI's ChatGPT. These can be represented by higher-order Markov chains whose kernels are learned during training. I will discuss the key points of its proof and put the result into perspective to scenarios relevant to natural language processing.