Hammersley's harness process: invariant distributions and height fluctuations

TL;DR

This paper analyzes the invariant distributions and height fluctuations of Hammersley's harness process, revealing universal behaviors and specific Gaussian cases, with implications for understanding stochastic heat equations.

Contribution

It characterizes invariant distributions in all dimensions and establishes height fluctuation behavior in one dimension as following the stochastic heat equation.

Findings

Unique invariant distribution under mild assumptions

Height fluctuations follow the stochastic heat equation in 1D

Process-level tightness in space-time trajectories

Abstract

We study the invariant distributions of Hammersley's serial harness process in all dimensions and height fluctuations in one dimension. Subject to mild moment assumptions there is essentially one unique invariant distribution, and all other invariant distributions are obtained by adding harmonic functions of the averaging kernel. We identify one Gaussian case where the invariant distribution is i.i.d. Height fluctuations in one dimension obey the stochastic heat equation with additive noise (Edwards-Wilkinson universality). We prove this for correlated initial data subject to polynomial decay of strong mixing coefficients, including process-level tightness in the Skorohod space of space-time trajectories.