Statistical Mechanics of the Chinese Restaurant Process: lack of self-averaging, anomalous finite-size effects and condensation

TL;DR

This paper analyzes the statistical mechanics of the Chinese Restaurant Process, revealing its condensation phenomenon, lack of self-averaging, and unique finite-size effects, which differ from traditional models.

Contribution

It provides a detailed statistical mechanics perspective on the Chinese Restaurant Process, highlighting its condensation and non-self-averaging properties with finite-size effects.

Findings

The process exhibits inevitable condensation with a finite number of dominant classes.

Finite-size effects lead to realization-dependent cutoffs and distribution behaviors.

The process lacks stationary state and self-averaging, unlike other models.

Abstract

The Pitman-Yor, or Chinese Restaurant Process, is a stochastic process that generates distributions following a power-law with exponents lower than two, as found in a numerous physical, biological, technological and social systems. We discuss its rich behavior with the tools and viewpoint of statistical mechanics. We show that this process invariably gives rise to a condensation, i.e. a distribution dominated by a finite number of classes. We also evaluate thoroughly the finite-size effects, finding that the lack of stationary state and self-averaging of the process creates realization-dependent cutoffs and behavior of the distributions with no equivalent in other statistical mechanical models.