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Mathematics > Probability

Title:Upper large deviations for maximal flows through a tilted cylinder

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 $n$.