Renewal stochastic processes with correlated events. Phase transitions along time evolution

TL;DR

This paper explores renewal stochastic processes with correlated events, revealing phase transitions over time, and introduces a statistical-mechanical framework to analyze different regimes of event clustering and independence.

Contribution

It develops a novel statistical-mechanical approach to renewal processes with correlated events, identifying phase transitions and linking them to large deviations theory.

Findings

Identification of second-order phase transitions in event clustering

Demonstration of regimes of correlated and independent events

Application of the framework to random walk processes

Abstract

We consider renewal stochastic processes generated by non-independent events from the perspective that their basic distribution and associated generating functions obey the statistical-mechanical structure of systems with interacting degrees of freedom. Based on this fact we look briefly into the less known case of processes that display phase transitions along time. When the density distribution \psi_{n}(t) for the occurrence of the n-th event at time t is considered to be a partition function, of a 'microcanonical' type for n 'degrees of freedom' at fixed 'energy' t, one obtains a set of four partition functions of which that for the generating function variable z and Laplace transform variable \epsilon, conjugate to n and t, respectively, plays a central role. These partition functions relate to each other in the customary way and in accordance to the precepts of large deviations theory, while the entropy, or Massieu potential, derived from \psi_{n}(t) satisfies an Euler relation. We illustrate this scheme first for an ordinary renewal process of events generated by a simple exponential waiting time distribution \psi (t). Then we examine a process modelled after the so-called Hamiltonian Mean Field (HMF) model that is representative of agents that perform a repeated task with an associated outcome, such as an opinion poll. When a sequence of (many) events takes place in a sufficiently short time the process exhibits clustering of the outcome, but for larger times the process resembles that of independent events. The two regimes are separated by a sharp transition, technically of the second order. Finally we point out the existence of a similar scheme for random walk processes.