Small ball probability, Inverse theorems, and applications

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page

The polyanalytic Ginibre ensembles

We consider a polyanalytic generalization of the Ginibre ensemble. This models allowing higher Landau levels (the Ginibre ensemble corresponds to the lowest Landau level). We study the local behavior of this point process under blow-ups.Comment: 31 page

Wildfire-induced short-term changes in a small mammal community increase prevalence of a zoonotic pathogen?

Natural disturbances like droughts and fires are important determinants of wildlife community structure and are suggested to have important implications for prevalence of wildlife-borne pathogens. After a major wildfire affecting >1,600 ha of boreal forest in Sweden in 2006, we took the rare opportunity to study the short-term response (2007-2010 and 2015) of small mammal community structure, population dynamics, and prevalence of the Puumala orthohantavirus (PUUV) hosted by bank voles (Myodes glareolus). We performed snap-trapping in permanent trapping plots in clear-cuts (n = 3), unburnt reference forests (n = 7), and the fire area (n = 7) and surveyed vegetation and habitat structure. Small mammal species richness was low in all habitats (at maximum three species per trapping session), and the bank vole was the only small mammal species encountered in the fire area after the first postfire year. In autumns of years of peak rodent densities, the trapping index of bank voles was lowest in the fire area, and in two of three peak-density years, it was highest in clear-cuts. Age structure of bank voles varied among forest types with dominance of overwintered breeders in the fire area in the first postfire spring. PUUV infection probability in bank voles was positively related to vole age. Infection probability was highest in the fire area due to low habitat complexity in burnt forests, which possibly increased encounter rate among bank voles. Our results suggest that forest fires induce cascading effects, including fast recovery/recolonization of fire areas by generalists like bank voles, impoverished species richness of small mammals, and altered prevalence of a rodent-borne zoonotic pathogen. Our pilot study suggests high human infection risk upon encountering a bank vole in the fire area, however, with even higher overall risk in unburnt forests due to their higher vole numbers