Hidden temperature in the KMP model

TL;DR

This paper characterizes the invariant measure of the KMP model, showing it as a product of boundary conditions and independent exponential variables, and explores correlations and hydrodynamic limits in one dimension.

Contribution

It provides a rigorous description of the invariant measure for the KMP model and confirms a conjecture related to large deviations, including correlation bounds and hydrodynamic limits in one dimension.

Findings

Invariant measure expressed as a product of boundary conditions and exponential variables.

Confirmation of a conjecture on large deviations of the model.

Bounded correlations and hydrodynamic limit in one-dimensional case.

Abstract

In the Kipnis Marchioro Presutti (KMP) model a positive energy $\zeta_i$ is associated with each vertex $i$ of a finite graph with a boundary. When a Poisson clock rings at an edge $ij$ with energies $\zeta_i,\zeta_j$, those values are substituted by $U(\zeta_i+\zeta_j)$ and $(1-U)(\zeta_i+\zeta_j)$, respectively, where $U$ is a uniform random variable in $(0,1)$. A value $T_j\ge 0$ is fixed at each boundary vertex $j$. The dynamics is defined in such way that the resulting Markov process $\zeta(t)$, satisfies that $\zeta_j(t)$ is exponential with mean $T_j$, for each boundary vertex $j$, for all $t$. We show that the invariant measure is the distribution of a vector $\zeta$ with coordinates $\zeta_i=T_i X_i$, where $X_i$ are iid exponential$(1)$ random variables, the law of $T$ is the invariant measure for an opinion random averaging/gossip model with the same boundary conditions of $\zeta$, and the vectors $X$ and $T$ are independent. The result confirms a conjecture based on the large deviations of the model. When the graph is one-dimensional, we bound the correlations of the invariant measure and perform the hydrostatic limit. We show that the empirical measure of a configuration chosen with the invariant measure converges to the linear interpolation of the boundary values.