Stochastic Dynamics of Discrete Curves and Multi-type Exclusion Processes

TL;DR

This paper investigates the stochastic dynamics of discrete curves and multi-type exclusion processes, analyzing their continuous limits, steady states, and fluctuations with a focus on reaction-diffusion systems and large deviation principles.

Contribution

It provides a detailed analysis of non-reversible reaction-diffusion systems derived from discrete curves, linking discrete cycle coefficients to fluid limit equations and studying fluctuations.

Findings

Invariant measures often have non-Gibbs form in non-reversible cases

Steady states are characterized using tagged particles and cycle expansions

Large deviation functionals are derived and solved for reversible systems

Abstract

This study deals with continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves with various kinds of coding representations. These systems are essentially of a reaction-diffusion nature. In the non-reversible case, the invariant measure has generally a non Gibbs form. The corresponding steady-state regime is analyzed in detail with the help of a tagged particle and a state-graph cycle expansion of the probability currents. As a consequence, the constants appearing in Lotka-Volterraequations --which describe the fluid limits of stationary states-- can be traced back directly at the discrete level to tagged particles cycles coefficients. Current fluctuations are also studied and the Lagrangian is obtained by an iterative scheme. The related Hamilton-Jacobi equation, which leads to the large deviation functional, is analyzed and solved in the reversible case for the sake of checking.