Upper bounds on the non-random fluctuations in first passage percolation with low moment conditions

TL;DR

This paper establishes upper bounds on non-random fluctuations in first passage percolation models with low moment conditions, providing insights into variance bounds especially when the limit shape exhibits flat edges.

Contribution

It introduces new upper bounds on non-random fluctuations under minimal moment assumptions and applies these results to variance lower bounds in specific geometric configurations.

Findings

Upper bounds on non-random fluctuations are derived.

Application to variance lower bounds when the limit shape has flat edges.

Results hold under low moment conditions, including moments greater than 1.

Abstract

We consider first passage percolation with i.i.d. weights on edges of the d-dimensional cubic lattice. Under the assumptions that a weight is equal to zero with probability smaller than the critical probability of bond percolation in the d-dimensional cubic lattice, and has moments bigger than 1, we investigate upper bounds on the so-called non-random fluctuations of the model. In addition, we give an application of our result to a lower bound for variance of the first passage percolation in the case where the limit shape has flat edges.