Colloquium in probability

We estimate the arrival time of vertices in a uniform random recursive tree from its unlabeled structure. Using centrality-based rankings, we derive tail bounds for the relative estimation error that are uniform in the vertex and the tree size. For the ranking induced by Jordan centrality, the probability that the estimate exceeds the true arrival time by a factor $S$ decays on the order of $1/S$, while the probability that it is smaller than the true arrival time by a factor $1/S$ decays exponentially in $S$. We introduce a refined centrality measure whose overestimation probability decays on the order of $(\log S)/S^{2}$, at the cost of a heavier lower tail of order $1/S^{2}$. These results identify a tradeoff between upper- and lower-tail performance in arrival-time estimation. Joint work with Simon Briend and Joost Jorritsma

The cover time of a Markov chain is the first time at which every state has been visited at least once. In this talk, we consider random walks that jump to stationarity every L steps; or at rate 1/L, where L is a given parameter that may diverge. We will show that the order of the expected cover time is the same in both setups and study when the cover time is concentrated. Aldous proved, in 1991, that for reversible Markov chains, the cover time is concentrated around its expectation if and only if the maximal expected hitting time is of strictly smaller order than the maximal expected cover time. We will give a similar concentration criterion and show that the concentration of the cover time is equivalent in both setups under certain conditions. Joint work with Omer Angel, Jonathan Hermon and Pietro Lavino.

Graph alignment asks which vertex of a given graph corresponds to which vertex of a second, unlabelled graph. We formalise this question as a statistical inference problem: two random graphs are correlated through a latent vertex correspondence, and the goal is to recover this correspondence with high probability in polynomial time. In this talk, we introduce a model of correlated Erdős–Rényi graphs that may have different numbers of vertices as well as varying edge densities. In the sparse setting with constant average degrees, these graphs are locally tree-like. Our alignment algorithm compares local tree neighbourhoods, which leads to a tree correlation testing problem. The feasibility of this testing problem exhibits a sharp phase transition, which we quantify in our main result. This result is surprising in two ways. First, the density of one tree can compensate for the sparsity of the other. Second, the proof revolves around a diagonalisation formula for the likelihood ratio over the space of unlabelled trees.

Lattice gauge theories are probabilistic models from statistical mechanics, introduced in the 1970s as a discrete model of gauge theories from physics. From a probabilistic point of view, they can be seen as a higher-dimensional analogue of classical spin systems such as the Ising model: spins live on vertices and interact along edges in the Ising model; while spins live on edges and interact around plaquettes (2-cells) in lattice gauge theories. In this talk, we will introduce the $\mathbb{Z}_2$ lattice gauge theory on the lattice $\mathbb{Z}^m, ~ m \geq 2$, and discuss some of its main questions, such as phase transitions and correlation inequalities. We will then focus on Ursell functions, or connected correlation functions, which are higher-order analogues of covariance. While Shlosman's theorem (1986) shows that Ursell functions on spins in the Ising model have alternating signs, depending only on the number of spins considered, we will see that this picture breaks down for lattice gauge theory. In particular, at sufficiently low temperature and in dimension $m \geq 3$, one can prove that for any number of Wilson loops, there is a choice of Wilson loop observables whose Ursell function is positive.

The model of oscillating random walks introduced by Kempermann is one of the simplest example of a Markov chain with discontinuous statistics. We consider the situation when this chain converges, after proper rescaling, towards a skew Brownian motion. In the talk I will discuss the corresponding local limit theorem.

We study the mixing time of the simple random walk on the giant component of supercritical $d$-dimensional random geometric graphs generated by the unit intensity Poisson Point Process in a $d$-dimensional cube of volume $n$. With $r_g$ denoting the threshold for having a giant component, we show that for every $\epsilon>0$ and any $r\geq(1+\epsilon)r_g$, the mixing time of the giant component is with high probability $\Theta(n^{2/d}/r^2)$, thereby closing a gap in the literature. Our analysis also implies that the relaxation time is of the same order. Joint work with M. Kiwi and C. Martinez.

Given a simple graph G = (V, E) and a map l0 : V → {+1, −1}, the majority dynamics on G with initial assignment of states l0 is a process that begins on day 0, and for each t ≥ 0 produces a new assignment of states lt+1 where each vertex takes the state of the majority of its neighbours, and remains at its previous state in the case of a tie. Specifically, for each v ∈ V , lt+1(v) = ( +1 if P u∈N (v) lt(u) > 0, or P u∈N (v) lt(u) = 0 and lt(v) = +1, −1 otherwise. (1) This process is a model for opinion exchange dynamics, with applications in many areas, such as politics, sociology, biophysics. While there exist results about the process even- tually reaching a 2-periodic stable state on all graphs, a natural question to study would be under which initial conditions is unanimity reached, and how quickly. When considering the question specifically in the case of Binomial Random Graphs, a longstanding conjecture, due to Benjamini, Chan, O’Donnel, Tamuz and Tan (2016) is the following: Conjecture. Let G ∼ G(n, p) be the binomial random graph with p = ω(1/n) and l0(v) be sampled uniformly at random from {+1, −1} for each v ∈ V . Then w.h.p. the majority dynamics process reaches unanimity after sufficiently many steps t. Steps towards proving the conjecture have been taken taken by gradually improving the range densities d = np for which the conjecture is known to be true, with the current best bound being d ≫ n1/3 log2/3 n due to Kim and Tran. Our work aims to prove the conjecture for the range n1/4 ≪ d ≤ O(n1/3), and lays the groundwork for proving the conjecture in the general case n1/(k+1) ≪ d ≤ O(n1/k). This is based on joint work with Nikolaos Fountoulakis, University of Birmingham.

Let $ (X_n)$ be a Markov chain and let $L_n$ denote its empirical measure at time $n$. We are interested in the large deviations of $(L_n)$. Roughly speaking, proving a large deviation principle for $(L_n)$ means proving that, for any given measure $\mu$, the probability of $L_n$ being close to $\mu$ decays with $n$ at an exponential rate (depending on $\mu$). The large deviations of $(L_n)$ have been studied since the 1970s and are well understood in "good" cases, in particular under assumptions of irreducibility of the Markov chain. However, very some simple Markov chains fail to satisfy these irreducibility assumptions. It turns out that transient states may play a role in large deviations, and complex behaviours can emerge at the large deviations scale when the Markov chain is not irreducible. I will describe these behaviours and present a new method for deriving the weak large deviation principle for $(L_n)$ in the reducible case, despite the resulting complication.

Optimal transport has become a central tool for comparing probability measures and extracting representative distributions from heterogeneous data — yet in many applications the objects of interest are stochastic processes, and the classical framework ignores a key structural feature: time and information. Indeed, classical Wasserstein barycenters ignore the filtration structure, making them ill-suited for problems in mathematical finance, stochastic control, and sequential decision-making. We study Fréchet means of Gaussian process laws in adapted Wasserstein space, where transport plans must respect the temporal flow of information. We prove that barycenters of Gaussian inputs exist, are Gaussian, and are unique. The key insight is a decomposition of the adapted Bures–Wasserstein distance into independent classical Bures–Wasserstein problems, one per time step, which yields both a clean characterization of the barycenter and a tractable fixed-point algorithm for its computation. Finally, we briefly discuss possible applications in robust stress testing of financial models and illustrate with numerical examples Wasserstein barycenters of autoregressive models. Based on joint work with Johannes Wiesel.

In this talk, we study a self-similar fragmentation process with index a>0, modeling the evolution of particles that break into smaller fragments over time. In this setting, the fragmentation rate of a particle of size u is proportional to u^a. We present asymptotic results describing the size of the largest fragment in the system. In particular, we establish a precise connection between the asymptotic behavior of the largest fragment and that of the so-called dislocation measure, which governs the underlying fragmentation mechanism. This is based on joint work with Samuel G. G. Johnston, Sandra Palau, and Joscha Prochno: https://arxiv.org/pdf/2409.11795

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).