The Annals of Probability

A stochastic telegraph equation from the six-vertex model

Alexei Borodin and Vadim Gorin

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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.

Article information

Ann. Probab., Volume 47, Number 6 (2019), 4137-4194.

Received: July 2018
Revised: February 2019
First available in Project Euclid: 2 December 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60G60: Random fields 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Six-vertex model telegraph equation Gaussian fields


Borodin, Alexei; Gorin, Vadim. A stochastic telegraph equation from the six-vertex model. Ann. Probab. 47 (2019), no. 6, 4137--4194. doi:10.1214/19-AOP1356.

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