The Annals of Probability

A stochastic telegraph equation from the six-vertex model

Alexei Borodin and Vadim Gorin

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1575277350

Digital Object Identifier
doi:10.1214/19-AOP1356

Mathematical Reviews number (MathSciNet)
MR4038051

Subjects
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

Keywords
Six-vertex model telegraph equation Gaussian fields

Citation

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. https://projecteuclid.org/euclid.aop/1575277350


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