## Distance evolutions in growing preferential attachment graphs.

Abstract: In this talk we will study the evolution of the graph distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with parameters such that the asymptotic degree distribution has infinite second moment. First, we grow the graph until it contains $t$ vertices, then we sample $u_t, v_t$ uniformly at random from the largest component and study the evolution of the graph distance as the surrounding graph grows. This yields a stochastic process in $t’\ge t$ that we call the distance evolution. We...

Read More## A limit law for the most favorite point of a simple random walk on a regular tree.

We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. We prove that, for the walk started from a leaf vertex and stopped upon hitting the root, as the depth of the tree tends to infinity the maximal time spent at any leaf converges, under suitable scaling and centering, to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale-like object associated with the square-root local-time process on the tree.

Read More## On some interacting particle systems related to random matrices.

Abstract: The talk is about two types of interacting particle systems related to the classical ensembles of random matrices. The prototypical examples are Dyson Brownian motion (also called non-intersecting Brownian motions) and Brownian TASEP (also called Brownian motions with one-sided collisions/reflections) respectively. I will discuss explicit formulae for their distributions, their correlation functions and some non-trivial connections between the two types of interacting particle systems. The bulk of the talk will be mainly focussed on surveying results for the Gaussian/Brownian...

Read More## The Wasserstein-Martingale projection of a Brownian motion given initial and terminal marginals.

Abstract: In one of its dynamic formulations, the optimal transport problem asks to determine the stochastic process that interpolates between given initial and terminal marginals and is as close as possible to the constant-speed particle. Typically, the answer to this question is a stochastic process with constant-speed trajectories. We explore the analogue problem in the setting of martingales, and ask: what is the martingale that interpolates between given initial and terminal marginals and is as close as possible to the constant volatility particle? The answer this time is a process...

Read More## The contact process with fitness on Galton-Watson trees.

Abstract: The contact process is a simple model for the spread of an infection in a structured population. We consider a variant of the contact process, where vertices are equipped with a random fitness representing inhomogeneities among individuals. In this inhomogeneous contact process, the infection is passed along an edge with rate proportional to the product of the fitness values of the vertices on either end. We assume that the underlying population structure is given by a Galton-Watson tree. Recent works by Huang/Durrett and Bhamidi et al have given necessary and sufficient...

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