Negative-weight percolation

TL;DR

This paper introduces a novel percolation problem involving negative weights on lattice edges, studying its transition behavior and critical exponents, revealing a different universality class from traditional percolation.

Contribution

It develops a new framework for negative-weight percolation, applying sophisticated algorithms to analyze its critical properties and universality class.

Findings

Negative-weight percolation has a distinct universality class from conventional percolation.

The transition exhibits unique critical exponents on various lattices.

Potential relation to spin-glass transitions in two dimensions.

Abstract

We describe a percolation problem on lattices (graphs, networks), with edge weights drawn from disorder distributions that allow for weights (or distances) of either sign, i.e. including negative weights. We are interested whether there are spanning paths or loops of total negative weight. This kind of percolation problem is fundamentally different from conventional percolation problems, e.g. it does not exhibit transitivity, hence no simple definition of clusters, and several spanning paths/loops might coexist in the percolation regime at the same time. Furthermore, to study this percolation problem numerically, one has to perform a non-trivial transformation of the original graph and apply sophisticated matching algorithms. Using this approach, we study the corresponding percolation transitions on large square, hexagonal and cubic lattices for two types of disorder distributions and determine the critical exponents. The results show that negative-weight percolation is in a different universality class compared to conventional bond/site percolation. On the other hand, negative-weight percolation seems to be related to the ferromagnet/spin-glass transition of random-bond Ising systems, at least in two dimensions.