# Power Lower Bounds for the Central Moments of the Last Passage Time for Directed Percolation in a Thin Rectangle

## TL;DR

This paper establishes lower bounds on the central moments of the last passage time in directed percolation models on thin rectangular grids, revealing how these moments scale with grid size for certain parameters.

## Contribution

It provides the first rigorous lower bounds on the moments of the last passage time in directed percolation within thin rectangles, extending understanding of fluctuation behavior.

## Key findings

- Lower bounds on the $r$-th central moment scale as $n^{r(1-eta)/2}$ for $0<eta<1/3$.
- Results apply to i.i.d. weights with finite support on $n imes n^eta$ grids.
- The bounds hold for sufficiently large $n$ and for all $r
eq 0$.

## Abstract

In directed last passage site percolation with i.i.d.~random weights with finite support over a $n\times\lfloor n^{\alpha}\rfloor$ grid, we prove that for $n$ large enough, the order of the $r$-th central moment, $1\le r<+\infty$, of the last passage time is lower bounded by $n^{r(1-\alpha)/2}$, $0<\alpha<1/3$.