Vol. 1, No. 1, 2020
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Joint distribution of Busemann functions in the exactly solvable corner growth model

Wai-Tong Louis Fan and Timo Seppäläinen
Vol. 1 (2020), No. 1, 55–100
DOI: 10.2140/pmp.2020.1.55
Abstract

The 1+1-dimensional corner growth model with exponential weights is a centrally important exactly solvable model in the Kardar–Parisi–Zhang class of statistical mechanical models. While significant progress has been made on the fluctuations of the growing random shape, understanding of the optimal paths, or geodesics, is less developed. The Busemann function is a useful analytical tool for studying geodesics. We describe the joint distribution of the Busemann functions, simultaneously in all directions of growth. As applications of this description we derive a marked point process representation for the Busemann function across a single lattice edge and calculate some marginal distributions of Busemann functions and semi-infinite geodesics.

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Keywords
Busemann function, Catalan number, Catalan triangle, cocycle, competition interface, corner growth model, directed percolation, geodesic, last-passage percolation, M/M/1 queue, multiclass fixed point, rhobar distance, TASEP
Mathematical Subject Classification 2010
Primary: 60K35, 60K37
Milestones
Received: 30 September 2019
Revised: 21 April 2020
Accepted: 11 May 2020
Published: 16 November 2020
Authors
Wai-Tong Louis Fan
Mathematics
Indiana University
Bloomington, IN
United States
Timo Seppäläinen
Department of Mathematics
University of Wisconsin-Madison
United States