Zero Range Condensation at Criticality

TL;DR

This paper analyzes the critical behavior of zero-range processes at the condensation transition, detailing how the maximum occupation number and fluctuations change at the critical density, providing a comprehensive understanding of the phase crossover.

Contribution

It establishes the law of large numbers and distributional limits for the maximum occupation number at criticality, revealing the transition from standard extreme value to Gaussian fluctuations.

Findings

Maximum occupation number jumps from zero to positive at a critical scale.

Fluctuations switch from extreme value to Gaussian at the critical point.

Mass outside the maximum is evenly distributed in the bulk.

Abstract

Zero-range processes with decreasing jump rates exhibit a condensation transition, where a positive fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number after adding or subtracting a subextensive excess mass of particles at the critical density. We establish a law of large numbers for the excess mass fraction in the maximum, which turns out to jump from zero to a positive value at a critical scale. Our results also include distributional limits for the fluctuations of the maximum, which change from standard extreme value statistics to Gaussian when the density crosses the critical point. Fluctuations in the bulk are also covered, showing that the mass outside the maximum is distributed homogeneously. In summary, we identify the detailed behaviour at the critical scale including sub-leading terms, which provides a full understanding of the crossover from sub- to supercritical behaviour.