Random surface growth with a wall and Plancherel measures for O(infinity)

TL;DR

This paper analyzes a Markov process of lozenge tilings with a reflecting boundary, revealing asymptotic behaviors, phase regions, and new determinantal processes, with connections to Plancherel measures for the infinite orthogonal group.

Contribution

It introduces a new model of surface growth with a wall and derives explicit correlation functions using representation theory.

Findings

Identification of frozen and liquid regions in the tiling model

Convergence to translation-invariant Gibbs measures in the liquid region

Derivation of new discrete Jacobi and symmetric Pearcey processes near the wall

Abstract

We consider a Markov evolution of lozenge tilings of a quarter-plane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall. We observe frozen and liquid regions, prove convergence of the local correlations to translation-invariant Gibbs measures in the liquid region, and obtain new discrete Jacobi and symmetric Pearcey determinantal point processes near the wall. The model can be viewed as the one-parameter family of Plancherel measures for the infinite-dimensional orthogonal group, and we use this interpretation to derive the determinantal formula for the correlation functions at any finite time moment.