Left-right crossings in the Miller-Abrahams random resistor network on a Poisson point process
Year: 2019
Type: preprint
Abstract: We consider the Miller-Abrahams (MA) random resistor network built on ahomogeneous Poisson point process (PPP) on $\mathbb{R}^d$, $d\geq 2$. Points of
the PPP are marked by i.i.d. random variables and the MA random resistor
network is obtained by plugging an electrical filament between any pair of
distinct points in the PPP. The conductivity of the filament between two points
decays exponentially in their distance and depends on their marks in a suitable
form prescribed by electron transport in amorphous materials. The graph
obtained by keeping filaments with conductivity lower bounded by a threshold
$\vartheta$ exhibits a phase transition at some $\vartheta_{\rm crit}$. Under
the assumption that the marks are nonnegative (or nonpositive) and bounded, we
show that in the supercritical phase the maximal number of vertex-disjoint
left-right crossings in a box of size $n$ is lower bounded by $Cn^{d-1}$ apart
an event of exponentially small probability. This result is one of the main
ingredients entering in the proof of Mott's law in [4].
Source: arXiv (Cornell University)
Authors: Alessandra Faggionato, Hlafo Alfie Mimun
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2995802452