Extreme value theory for constrained physical systems

Abstract

We investigate extreme value theory for physical systems with a global conservation law which describe renewal processes, mass transport models and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a non-analytical point in the middle of the support. We expose exact relationships between constrained extreme value theory and well-known quantities of the underlying stochastic dynamics, all valid beyond the midpoint in generality, i.e. even far from the thermodynamic limit. For example for renewal processes, the distribution of the maximum time between two renewal events is exactly related to the mean number of these events. In the thermodynamic limit, we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory. For example for the renewal process, we unravel dual scaling of the extreme value distribution, pointing out two types of limiting laws: a normalisable scaling function for the typical statistics and a non-normalised state describing the rare events.