Mott's law for the critical conductance of Miller-Abrahams random resistor network

TL;DR

This paper derives Mott's law for the critical conductance in a Miller-Abrahams resistor network on a Poisson point process, providing a percolative characterization and supporting its universality, with implications for understanding electrical conduction.

Contribution

It introduces a rigorous derivation of Mott's law for the critical conductance in a random resistor network and offers a percolative perspective on the key factors involved.

Findings

Derived Mott's law for the critical conductance

Provided a percolative characterization of the temperature-dependent factor

Supported the universality of the law through mathematical arguments

Abstract

In this short note we derive Mott's law for the critical conductance of the Miller-Abrahams random resistor network on a Poisson point process on $\mathbb{R}^d$, $d\geq 2$, and we give a percolative characterization of the factor preceding the temperature dependent term $\beta^\frac{\alpha+1}{\alpha+1+d} $. We also give mathematical arguments supporting its universality. This note is a preliminary version of a more extended work, where we also discuss the equality between the effective conductance of the resistor network and the critical conductance.