1 Introduction

This project formalizes the convergence properties of Last Passage Percolation (LPP) models with approximate Gumbel distributions. We establish that LPP with \(N\)-approximate Gumbel weights converges to the same GUE Tracy-Widom distribution as exact Gumbel LPP, provided that \(N\) grows appropriately with the grid size.

The main result uses a coupling argument between exact and approximate Gumbel distributions, combined with perturbation bounds to control the difference between the two models.