Condensation and Extreme Value Statistics

TL;DR

This paper investigates condensation phenomena in mass transport models by analyzing the extreme value statistics of the largest mass, revealing distinct distribution behaviors at different density regimes.

Contribution

It links condensation transition to extreme value statistics, providing a new perspective on the behavior of the largest mass in such models.

Findings

At subcritical densities, the largest mass distribution follows Gumbel statistics.

At the critical density, the distribution transitions to Fréchet.

Above the critical density, a different distribution characterizes the largest mass.

Abstract

We study the factorised steady state of a general class of mass transport models in which mass, a conserved quantity, is transferred stochastically between sites. Condensation in such models is exhibited when above a critical mass density the marginal distribution for the mass at a single site develops a bump, $p_{\rm cond}(m)$, at large mass $m$. This bump corresponds to a condensate site carrying a finite fraction of the mass in the system. Here, we study the condensation transition from a different aspect, that of extreme value statistics. We consider the cumulative distribution of the largest mass in the system and compute its asymptotic behaviour. We show 3 distinct behaviours: at subcritical densities the distribution is Gumbel; at the critical density the distribution is Fr\'echet, and above the critical density a different distribution emerges. We relate $p_{\rm cond}(m)$ to the probability density of the largest mass in the system.