Permutation invariance in last-passage percolation and the distribution of the Busemann process
5 affiliations
- 1Department of Mathematics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC 27695-8205 USA
- 2School of Mathematics and Statistics, University College Dublin, Dublin, Ireland
- 3Department of Statistics, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB UK
- 4Mathematics Department, University of Wisconsin–Madison, Van Vleck Hall, 480 Lincoln Dr., Madison, WI 53706-1388 USA
- 5Department of Mathematics, Columbia University, Room 624, MC 4432, 2990 Broadway, New York, NY 10027 USA
Abstract
In i.i.d. exponential last-passage percolation, we describe the joint distribution of Busemann functions, over all edges and over all directions, in terms of a joint last-passage problem in a finite inhomogeneous environment. More specifically, the Busemann increments within a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}k× ℓ\end{document}k×ℓ grid, and associated to d different directions, are equal in distribution to a particular collection of last-passage increments inside a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}(k+d−1)× (ℓ +d−1)\end{document}(k+d-1)×(ℓ+d-1) grid. The joint Busemann distribution was previously described along a horizontal line by Fan and the fourth author, using certain queueing maps. By contrast, our new description explicitly gives the joint distribution for any collection of edges (not just along a horizontal line) using only finitely many random variables. Our result thus provides an exact and accessible way to sample from the joint distribution. In the proof, we rely on one-directional marginal distributions of the inhomogeneous Busemann functions recently studied by Janjigian and the second and fourth authors. The second ingredient of our proof is a novel joint invariance of inhomogeneous last-passage times under permutations of the inhomogeneity parameters. Our proof of the invariance is different from earlier proofs of such results, using the Burke property instead of the RSK correspondence, and leading to an explicit coupling of the weights before and after the permutation of the parameters.
Related
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- Conditional Flat Exponential Directed Last Passage Percolation under a One-Point Upper Large Deviation Event
BibTeX
@article{bates2026permutation,
title = {{Permutation invariance in last-passage percolation and the distribution of the Busemann process}},
author = {Erik Bates and Elnur Emrah and James Martin and Timo Seppäläinen and Evan Sorensen},
journal = {Probability Theory and Related Fields},
year = {2026},
month = jul,
doi = {10.1007/s00440-026-01516-7},
url = {https://doi.org/10.1007/s00440-026-01516-7},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Permutation invariance in last-passage percolation and the distribution of the Busemann process
AU - Erik Bates
AU - Elnur Emrah
AU - James Martin
AU - Timo Seppäläinen
AU - Evan Sorensen
PY - 2026
DA - 2026/07/22
JO - Probability Theory and Related Fields
DO - 10.1007/s00440-026-01516-7
UR - https://doi.org/10.1007/s00440-026-01516-7
AB - In i.i.d. exponential last-passage percolation, we describe the joint distribution of Busemann functions, over all edges and over all directions, in terms of a joint last-passage problem in a finite inhomogeneous environment. More specifically, the Busemann increments within a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}k× ℓ\end{document}k×ℓ grid, and associated to d different directions, are equal in distribution to a particular collection of last-passage increments inside a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}(k+d−1)× (ℓ +d−1)\end{document}(k+d-1)×(ℓ+d-1) grid. The joint Busemann distribution was previously described along a horizontal line by Fan and the fourth author, using certain queueing maps. By contrast, our new description explicitly gives the joint distribution for any collection of edges (not just along a horizontal line) using only finitely many random variables. Our result thus provides an exact and accessible way to sample from the joint distribution. In the proof, we rely on one-directional marginal distributions of the inhomogeneous Busemann functions recently studied by Janjigian and the second and fourth authors. The second ingredient of our proof is a novel joint invariance of inhomogeneous last-passage times under permutations of the inhomogeneity parameters. Our proof of the invariance is different from earlier proofs of such results, using the Burke property instead of the RSK correspondence, and leading to an explicit coupling of the weights before and after the permutation of the parameters.
SN - 0178-8051
SN - 1432-2064
ER -