A stochastic telegraph equation from the six-vertex model

TL;DR

This paper introduces a stochastic telegraph equation with white noise inhomogeneity, showing that its solutions are Gaussian fields that describe fluctuations in the height function of the stochastic six-vertex model, linking stochastic PDEs to statistical mechanics.

Contribution

It defines a new stochastic telegraph equation with white noise inhomogeneity and connects its solutions to the asymptotic fluctuations of the six-vertex model's height function.

Findings

Solutions are two-dimensional Gaussian fields.

Fluctuations of the height function follow the stochastic telegraph equation.

Limit shape governed by the deterministic telegraph equation.

Abstract

A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six vertex model in a quadrant. The corresponding law of large numbers -- the limit shape of the height function -- is described by the (deterministic) homogeneous telegraph equation.