Upper large deviations for maximal flows through a tilted cylinder

TL;DR

This paper investigates the large deviation probabilities for maximal flows in a tilted cylinder within a first passage percolation model, revealing how decay rates depend on edge capacity distributions and cylinder geometry.

Contribution

It provides new large deviation estimates for maximal flows in tilted cylinders, highlighting the influence of capacity tail behavior and geometric parameters.

Findings

Decay rates depend on the tail of edge capacity distribution.

Exponential decay with the volume of the cylinder when capacities have exponential moments.

Different regimes of decay depending on the height function h(n).

Abstract

We consider the standard first passage percolation model in $\ZZ^d$ for $d\geq 2$ and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to $n$ and whose height is $h(n)$ for a certain height function $h$. We denote this maximal flow by $\tau_n$ (respectively $\phi_n$). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than $\nu(\vec{v})+\eps$ for some positive $\eps$, where $\nu(\vec{v})$ is the almost sure limit of the rescaled variable $\tau_n$ when $n$ goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable $\tau_n$ depends on the tail of the distribution of the capacities of the edges: it can decays exponentially fast with $n^{d-1}$, or with $n^{d-1} \min(n,h(n))$, or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable $\phi_n$ decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that $\nu(\vec{v})$ is not in general the almost sure limit of the rescaled maximal flow $\phi_n$, but it is the case at least when the height $h(n)$ of the cylinder is negligible compared to $n$.