Noela Müller (Eindhoven) - Sharp thresholds and hitting times for factors in the Binomial random graph

Let F be a graph on r vertices and let G be a graph on n vertices. Then an F-factor in G is a subgraph of G composed of n/r vertex-disjoint copies of F, if r divides n. In other words, an F-factor yields a partition of the n vertices of G. The study of such F-factors in the Erdős–Rényi random graph dates back to Erdős himself. Decades later, in 2008, Johansson, Kahn and Vu established the thresholds for the existence of an F-factor for strictly 1-balanced F – up to the leading constant. The sharp thresholds, meaning the leading constants, were obtained only
recently by Riordan and Heckel, but only for complete graphs and for so-called ‘nice’ graphs. Their results rely on sophisticated couplings that utilize the recent, celebrated solution of Shamir’s problem by Kahn.

The talk will explain the basic ideas underlying the couplings by Riordan and Heckel, as well as extensions to strictly 1-balanced graphs F and hitting time results.

The talk is based on joint work with Fabian Burghart, Annika Heckel, Marc Kaufmann and Matija Pasch.

​Malwina Luczak (Manchester) - Cutoff for the logistic SIS epidemic model with self-infection

We study a variant of the classical Markovian logistic SIS epidemic model on a complete graph, which has the additional feature that healthy individuals can become infected without contacting an infected member of the population. This additional ``self-infection'' is used to model situations where there is an unknown source of infection or an external disease reservoir, such as an animal carrier population. In contrast to the classical logistic SIS epidemic model, the version with self-infection has a non-degenerate stationary distribution, and we derive precise asymptotics for the time to converge to stationarity (mixing time) as the population size becomes large. It turns out that the chain exhibits the cutoff phenomenon, which is a sharp transition in time from one to zero of the total variation distance to stationarity. We obtain the exact leading constant for the cutoff time, and show the window size is constant (optimal) order.

We further place this result within the context of a recent more general cutoff result.

This is joint work with Roxanne He and Nathan Ross, and the more general cutoff result is joint work with Barbour and Brightwell.