## Competition processes on hyperbolic non-amenable graphs.

Abstract: We consider two first-passage percolation processes, FPP_1 and FPP_\lambda, spreading with rates 1 and \lambda respectively, on a graph G with bounded degree. FPP_1 starts from a single source, while the initial configuration of FPP_\lambda consists of countably many seeds distributed according to a product of iid Bernoulli random variables of parameter \mu on the set of vertices. This model is known as “First passage percolation in a hostile environment” (FPPHE), it was introduced by Stauffer and Sidoravicius as an auxiliary model for investigating a notoriously...

Read More## Quantitative hydrodynamic limit and regularity for Langevin dynamics for gradient interface models.

Abstract: Lanngevin dynamics for gradient interface models are important in statistical physics due to their connection with random surfaces. It is of particular interest to understand their behavior over large-scales. In this direction a number of results have been established in the last 20 years (including the hydrodynamic limit of Funaki-Spohn and the scaling limit of Naddaf-Spencer and Giacomin-Olla-Spohn). In this talk, we will present the model, its motivations and main results. We will study a connection with the stochastic homogenization of nonlinear equations, discuss some...

Read More## Tail bounds for detection times in mobile hyperbolic graphs.

Abstract: Motivated by Krioukov et al.’s model of random hyperbolic graphs for real-world networks, and inspired by the analysis of a dynamic model of graphs in Euclidean space by Peres et al., we introduce a dynamic model of hyperbolic graphs in which vertices are allowed to move according to a Brownian motion maintaining the distribution of vertices in hyperbolic space invariant. For different parameters of the speed of angular and radial motion, we analyze tail bounds for detection times of a fixed target and obtain a complete picture, for very different regimes, of how and when the...

Read More## A simple model for an epidemic with contact tracing and cluster isolation, and a detection paradox.

Abstract: We determine the distributions of some random variables related to a simple model of an epidemic with contact tracing and cluster isolation, which is inspired by a recent work of Bansaye, Gu and Yuan. Notably, we compute explicitly the asymptotic proportion of isolated clusters with a given size amongst all isolated clusters, conditionally on survival of the epidemic. Somewhat surprisingly, the latter differs from the distribution of the size of a typical cluster at the time of its detection; and we explain the reasons behind this seeming paradox.

Read More## Quantum Decoherence for probabilists.

Resumen: Quantum decoherence (QD) is today one cornerstone in the development of quantum computing (QC). This refers to the collapse of a quantum state into a classical one. From a mathematical point of view, its modelling has also been a major problem, motivating the development of new research in open systems theory. One could classify today this phenomenon at the interface between non-commutative and commutative probabilities. The general question is: how a quantum evolution becomes classical? Is this inevitable? Shall QC live with that? The talk will provide a panorama on the...

Read More## Rough walks in random environments.

Resumen: We shall discuss functional CLTs for additive functionals of Markov processes and regenerative processes lifted to the rough path space. The limiting rough path has two levels of which the first one is a Brownian motion with a well-known covariance matrix. However, in the second level we see a new feature: it is the iterated integral of the same Brownian motion perturbed by a deterministic linear function called the area anomaly and characterized in terms of the model. With that one obtains sharper information on the limiting path. The construction of new examples for SDE...

Read More