Multiple phases in stochastic dynamics: geometry and probabilities

TL;DR

This paper introduces a geometric approach to stochastic dynamics using eigenvectors of transition matrices, revealing phase transitions and hierarchical structures, and enabling probability calculations of system states.

Contribution

It presents the observable-representation of state space, a novel geometric method to analyze phases and hierarchical structures in stochastic systems.

Findings

Reveals phases as extremal points in the geometric representation

Allows hierarchical structure observation in stochastic dynamics

Provides a method to calculate transition probabilities to asymptotic states

Abstract

Stochastic dynamics is generated by a matrix of transition probabilities. Certain eigenvectors of this matrix provide observables, and when these are plotted in the appropriate multi-dimensional space the phases (in the sense of phase transitions) of the underlying system become manifest as extremal points. This geometrical construction, which we call an \textit{observable-representation of state space}, can allow hierarchical structure to be observed. It also provides a method for the calculation of the probability that an initial points ends in one or another asymptotic state.