Condensation and extremes for a fluctuating number of independent random variables

TL;DR

This paper unifies the theory of condensation and extremes across three classes of stochastic processes with either fixed or fluctuating numbers of components, providing new insights into their extreme value statistics and fluctuation mechanisms.

Contribution

It offers a unified framework for understanding condensation and extremes in various stochastic processes, including new insights into their extreme value statistics and fluctuations.

Findings

Unified presentation of condensation phenomena across different process classes

New insights into the extreme value statistics of these processes

Deeper understanding of the fluctuation mechanisms of the condensate

Abstract

We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the consideration of particular models and highlight the essentials. Besides giving a unified presentation of the theory, this work investigates facets so far unexplored in previous studies. Specifically, we show how the study of the class of random allocation models and zero-range processes can serve as a backdrop for the study of the two other classes of processes central to the present work -- tied-down and free renewal processes. We then present new insights on the extreme value statistics of these three classes of processes which allow a deeper understanding of the mechanism of condensation and the quantitative analysis of the fluctuations of the condensate.