Precise large deviations in geometric last passage percolation

TL;DR

This work obtains precise large-deviation asymptotics for the geometric last-passage percolation times $G_{n,m}$ by exploiting dualities with random-matrix ensembles. The core strategy translates tail events for $G_{n,m}$ into TUE moment problems and JUE large-deviation events, enabling full expansions including constant terms. The analysis combines Schur-function dualities, constrained equilibrium measures for JUE, and Hankel-determinant expansions to yield detailed lower- and upper-tail formulas across almost-square and rectangular geometries, with complementary results for CUE/Coulomb-gas settings. The results provide a unified framework linking LPP to non-Hermitian and Hermitian random-matrix theories and offer explicit asymptotics for moments of truncated-unitary characteristic polynomials and for constrained JUE distributions, contributing to the precise understanding of KPZ-related tail behaviors and Coulomb-gas free energies.

Abstract

We study the last passage time in geometric last passage percolation (LPP). As the system size increases, we derive precise large deviation probabilities -- up to and including the constant terms -- for both the lower and upper tails. A key step in proving these results is to establish a duality formula that reformulates the LPP problem in terms of the largest eigenvalue in the Jacobi unitary ensemble (JUE). In addition, we establish a second duality formula, which relates the LPP problem to the truncated unitary ensemble (TUE). Using this, we also derive asymptotics for the moments of the absolute value of characteristic polynomials of the TUE, which may be of independent interest.

Figures (6)

• Figure 1: Plots of the last passage paths (red lines) from $(0,0)$ to $(n,m)$ with $q=1/\sqrt{2}$. The panels correspond to (A) $n=m=10$, (B) $n=m=20$, and (C) $n=m=40$. Lighter shading indicates smaller values of the weights $\omega_{i,j}$.
• Figure 2: The histogram was made by computing $10^{4}$ samples of $\frac{G_{N, N} - N \omega(\gamma, q)}{\sigma(\gamma, q) N^{1/3}}$ with $N=400$. The blue curve is the density $F_{\mathrm{TW}}'(x)$ of the Tracy-Widom distribution.
• Figure 3: In each plot, the points are the eigenvalues of a TUE matrix with the indicated values of $N$ and $M$. The thin curves are circles centred at $0$ of radii $1$ and $1/\sqrt{1+\rho}$.
• Figure 4: Histograms of the eigenvalues of a JUE matrix \ref{['def of JUE matrix']} of size $n = 1000$, for various values of $\alpha$ and $\beta$. The corresponding limiting densities \ref{['def of Wachter distribution']} are the solid curves.
• Figure 5: Plots show limiting densities of the constrained JUE. (A) and (B) correspond to the pulled regime of Proposition \ref{['Prop_constrained Wacther']}, and (C) and (D) to the pushed regime.

Theorems & Definitions (41)

• Theorem 1.1: Lower tail probability in an almost-square lattice
• Theorem 1.2: Lower tail probability in a rectangular lattice
• Remark 1
• Remark 2
• Theorem 1.3: Upper tail probability
• Remark 3: Comparison with Johansson's formula
• Theorem 2.1: Duality between LPP and TUE
• Theorem 2.2: Moments of the characteristic polynomial of TUE matrices at weak non-unitarity
• Remark 4
• Corollary 2.3: Moments of the characteristic polynomial of CUE