November 4, 2022
Location: UU Minnaert, 013
Planarity, percolation, and height functions
Abstract: Fröhlich and Spencer proved the Berezinskii-Kosterlitz-Thouless transition in 1981, through a relation with delocalisation of height functions. Their delocalisation proof goes through a relation with the Coulomb gas. In recent years it is becoming clear that this phase transition can also be understood in a simpler way through couplings with planar percolation models. The purpose of this talk is to provide a soft introduction into this topic and to present recent advancements.
Stochastic epidemics on inhomogenous random graphs with degree-dependent contact rates
We consider a stochastic SIR (Susceptible, Infectious, Recovered) epidemic on a large inhomogeneous random graph (as introduced by Bollobás, Janson and Riordan). In the first part of the presentation I will introduce some basic models for the spread of infectious diseases in structured populations modeled by inhomogeneous random graphs.
In most of the basic models for epidemics on networks the (possibly stochastic) processes describing the contacts between individuals follow the same law for every pair of neighbours and are therefore independent of the degrees of the individuals. This simplifying assumption ignores possible dependencies between the number of neighbours of an individual and the manner in which he or she interacts with those neighbours. For instance, an individual with many friends may have more interactions overall than an individual with few friends, but fewer interactions with each friend due to time constraints.
In the second part of the talk I will investigate the impact of the above assumption. To this end, we consider a new model where the infectious contact rate (to be explained in talk) between two neigbours depends on their degrees. We make assumptions that, heuristically, ensure that the per-neighbour contact rates of an individual decrease with the expected number of neighbours of that individual.
We investigate the basic reproduction number $R_0$, the probability of a large outbreak and (in case of a large outbreak) the asymptotic fraction of the population infected at the end of the epidemic.
This talk is based on joint work with Carolina Fransson (Stockholm University).