Precise large deviations in geometric last passage percolation

TL;DR

This paper derives precise large deviation probabilities for geometric last passage percolation, connecting the problem to eigenvalue distributions in random matrix ensembles and providing detailed asymptotics.

Contribution

It introduces duality formulas linking LPP to eigenvalue problems in Jacobi and truncated unitary ensembles, enabling exact large deviation analysis.

Findings

Derived large deviation probabilities including constant terms

Established duality formulas relating LPP to eigenvalue problems

Obtained asymptotics for moments of characteristic polynomials of TUE

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.