Phase transition in equilibrium fluctuations of symmetric slowed exclusion

TL;DR

This paper investigates how the presence of a slow bond affects the equilibrium fluctuations of density, current, and tagged particles in a symmetric exclusion process, revealing three distinct fluctuation regimes and explicit covariance formulas.

Contribution

It provides a detailed analysis of the fluctuation behavior in symmetric exclusion with a slow bond, including explicit covariance computations and a new family of Gaussian processes at criticality.

Findings

Three different fluctuation regimes depending on the parameter β.

Explicit covariance formulas for each regime.

A family of Gaussian processes interpolating fractional Brownian motion at criticality.

Abstract

We analyze the equilibrium fluctuations of the density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is $\alpha n^-\beta$, where $\alpha,\beta\geq{0}$ and $n$ is the scaling parameter. Depending on the regime of $\beta$, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value $\beta=1$, starting a tagged particle near the slow bond, we obtain a family of gaussian processes indexed in $\alpha$, interpolating a fractional brownian motion of Hurst exponent 1/4 and the degenerate process equal to zero.