Stick-Breaking Model for Variable-Range Hopping

TL;DR

This paper introduces a hierarchical stick-breaking model to analyze the optimal conduction path in one-dimensional variable-range hopping, aligning well with numerical percolation results and simplifying analysis.

Contribution

It presents a novel hierarchical stick-breaking approach to model variable-range hopping, providing a more analyzable framework that matches numerical findings.

Findings

Hierarchical model agrees with numerical percolation results.

Hopping length distribution is well approximated by stick-breaking.

Model simplifies analysis of conduction paths.

Abstract

We consider the optimal conduction path of the one-dimensional variable-range hopping problem. We describe a hierarchical procedure for constructing the path which is in excellent agreement with numerical results obtained from a percolation approach. The advantage of the hierarchical construction is that it is easier to analyse. We show that the distribution of hopping lengths is well approximated by a model for the repeated breaking of a stick at its weakest point, until the fragments are too strong to be broken.