Extreme value theory for constrained physical systems

Marc Höll, Wanli Wang, Eli Barkai

Published 2020-06-11Version 1

We investigate extreme value theory (EVT) of physical systems with a global conservation law which describe renewal processes, mass transport models and long-range interacting spin models. A special feature is that the distribution of the extreme value exhibits a non-analytical point in the middle of the support. We reveal three exact relationships between constrained EVT and random walks, density of condensation, and rate of renewals, all valid in generality beyond the mid point. For example for renewal processes at time $T$ describing blinking quantum dots, photon arrivals, zero crossings of Brownian motion and many other systems the cumulative distribution of the maximum $m$ is $F(m)=1-\varphi(m) \langle N(T-m) \rangle$ where $\langle N \rangle$ is the average number of jumps in the process and $\varphi$ is the survival probability. Our theory provides a general tool to describe the constrained EVT close and far from thermodynamic limit, and is therefore a unified extension of classical EVT.