On the Thermodynamic Limit in Random Resistors Networks

TL;DR

This paper investigates the behavior of random resistor networks on Euclidean lattices, proving the existence and independence of the thermodynamic limit of dissipation per volume under certain conditions.

Contribution

It formulates a variational principle for the model and establishes the finiteness and boundary condition independence of the thermodynamic limit.

Findings

Thermodynamic limit of dissipation per volume is finite almost surely.

Limit is independent of boundary conditions for specific cases.

Results hold in the mean as well as almost surely.

Abstract

We study a random resistors network model on a euclidean geometry $\bt{Z}^d$. We formulate the model in terms of a variational principle and show that, under appropriate boundary conditions, the thermodynamic limit of the dissipation per unit volume is finite almost surely and in the mean. Moreover, we show that for a particular thermodynamic limit the result is also independent of the boundary conditions.