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Last-passage percolation and product-matrix ensembles

Sergey Berezin, Eugene Strahov

TL;DR

This work analyzes directed last-passage percolation in a planar layered environment composed of blocks with exponential clocks, unveiling integrability through a Fredholm determinant tied to truncated-unitary product-matrix kernels. The authors establish a finite-dimensional determinant expression for the last-passage time process and prove a scaling limit to a continuous-time, multi-time critical stochastic process governed by an extended critical kernel. They further connect the critical kernel to hard-edge limits of Ginibre and truncated-unitary product-matrix ensembles, proving convergence to an extended hard-edge kernel and showing a hard-to-soft edge transition that recovers the extended critical kernel. The results fuse last-passage percolation with product-matrix determinantal point processes, providing a coherent framework that encompasses Schur processes, RSK combinatorics, and random-matrix theory, with potential implications for multi-time stochastic growth models and integrable probability.

Abstract

We introduce and study a model of directed last-passage percolation in planar layered environment. This environment is represented by an array of random exponential clocks arranged in blocks, for each block the average waiting times depend only on the local coordinates within the block. The last-passage time, the total time needed to travel from the source to the sink located in a given block, maximized over all the admissible paths, becomes a stochastic process indexed by the number of blocks in the array. We show that this model is integrable, particularly the probability law of the last-passage time process can be determined via a Fredholm determinant of the kernel that also appears in the study of products of random matrices. Further, we identify the scaling limit of the last-passage time process, as the sizes of the blocks become infinitely large and the average waiting times become infinitely small. Finite-dimensional convergence to the continuous-time critical stochastic process of random matrix theory is established.

Last-passage percolation and product-matrix ensembles

TL;DR

This work analyzes directed last-passage percolation in a planar layered environment composed of blocks with exponential clocks, unveiling integrability through a Fredholm determinant tied to truncated-unitary product-matrix kernels. The authors establish a finite-dimensional determinant expression for the last-passage time process and prove a scaling limit to a continuous-time, multi-time critical stochastic process governed by an extended critical kernel. They further connect the critical kernel to hard-edge limits of Ginibre and truncated-unitary product-matrix ensembles, proving convergence to an extended hard-edge kernel and showing a hard-to-soft edge transition that recovers the extended critical kernel. The results fuse last-passage percolation with product-matrix determinantal point processes, providing a coherent framework that encompasses Schur processes, RSK combinatorics, and random-matrix theory, with potential implications for multi-time stochastic growth models and integrable probability.

Abstract

We introduce and study a model of directed last-passage percolation in planar layered environment. This environment is represented by an array of random exponential clocks arranged in blocks, for each block the average waiting times depend only on the local coordinates within the block. The last-passage time, the total time needed to travel from the source to the sink located in a given block, maximized over all the admissible paths, becomes a stochastic process indexed by the number of blocks in the array. We show that this model is integrable, particularly the probability law of the last-passage time process can be determined via a Fredholm determinant of the kernel that also appears in the study of products of random matrices. Further, we identify the scaling limit of the last-passage time process, as the sizes of the blocks become infinitely large and the average waiting times become infinitely small. Finite-dimensional convergence to the continuous-time critical stochastic process of random matrix theory is established.

Paper Structure

This paper contains 19 sections, 23 theorems, 221 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Directed last passage percolation in layered environment
  4. Convergence to critical stochastic process
  5. Critical stochastic process and hard-edge limits of product-matrix ensembles
  6. Product of Ginibre matrices and its hard-edge scaling limit
  7. Product of truncated unitary matrices and its hard-edge scaling limit
  8. Transition to extended critical kernel
  9. Last passage percolation
  10. Young diagrams, tableaux, and integer arrays
  11. Arrays of geometric random variables and Schur point processes
  12. Arrays of exponential random variables and limits of Schur point processes
  13. Proof of Theorem \ref{['thm21']}
  14. Proofs of Theorems \ref{['thm22']} -- \ref{['thm25']}
  15. Auxiliary formulas and asymptotics
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $r_1, \ldots, r_q \in \mathbb{N}$ be pairwise distinct. Then, the finite-dimensional distributions of the stochastic process $\left(\mathfrak{T}(k),\ k \in \mathbb{N} \right)$ associated with the arrays eq:Aarray of the exponential random variables eq:exp_laws are given by where $s_k \in \mathbb{R}_{>0}$ and the kernel is given by where by definition $\nu_0=1$ and $\ell_0 = -n$, the contours

Figures (6)

  • Figure 1: An illustration of directed last-passage percolation in layered environment.
  • Figure 2: The contour $S_{\sigma}$ and $S_{\zeta}^{(n)}$.
  • Figure 3: The integration contours ${S}_\sigma$ and ${S}_{\zeta}$. The contour $S_\sigma$ crosses the real axis at $-c$, where $c>0$, and does not intersect $S_\zeta$.
  • Figure 4: The integration contours ${S}_\sigma$ and ${S}_{\zeta}$ in \ref{['eq:Meij_hardedge']}.The contour $S_\sigma$ crosses the real axis at $-c$, where $c>0$, and does not intersect $S_\zeta$. All the poles $\{ -\nu_0, -\nu_0-1, \ldots; -\nu_1, -\nu_1-1, \ldots\}$ in the $\sigma$-plane lie inside the domain bounded by $S_\sigma$.
  • Figure 5: The contour $S_{\sigma}$ and $S_{\zeta}^{(n)}$.
  • ...and 1 more figures

Theorems & Definitions (44)

  • Theorem 2.1
  • Remark 1
  • Remark 2
  • Theorem 2.2
  • Remark 3
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Lemma 3.2
  • ...and 34 more