A law of the iterated logarithm for directed last passage percolation

TL;DR

This paper proves a law of the iterated logarithm for directed last passage percolation, showing convergence to Tracy-Widom distribution with a novel loglog normalization, advancing understanding of fluctuation behaviors.

Contribution

It introduces a new law of the iterated logarithm for last passage times in directed percolation with exponential or geometric variables, using sharp tail bounds and superadditivity.

Findings

Establishes a law of the iterated logarithm with loglog normalization.

Shows convergence of scaled passage times to Tracy-Widom distribution.

Provides a weaker liminf result for the same model.

Abstract

A law of the iterated logarithm is established for the last passage times of directed percolation on rectangles in the plane over exponential or geometric independent random variables, rescaled to converge to the Tracy-Widom distribution. The normalization is of loglog type, in contrast with the log normalization for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result for the liminf is also discussed.