Condensation and extremes for a fluctuating number of independent random variables

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.

arXiv comment55 pages, 12 figures, final version, to be published in Journal of Statistical Physics

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• Discrete Sampling of Extreme Events Modifies Their Statistics
• Finite-size localization scenarios in condensation transitions
• Accurately approximating extreme value statistics
• On Random Allocation Models in the Thermodynamic Limit
• Fluctuation dominated phase ordering in coarse-grained depth models: Domain wall structures, extreme values and coarsening
• Partition function zeros of zeta-urns
• Yang-Lee zeros for real-space condensation
• Partition Function Zeros of Paths and Normalization Zeros of ASEPS
• Big jump principle for heavy-tailed random walks with correlated increments
• Mean-field theory of vector spin models on networks with arbitrary degree distributions