Mean-field behavior of the negative-weight percolation model on random regular graphs

TL;DR

This paper explores the mean-field properties of the negative-weight percolation model on random regular graphs through analytical and numerical methods, confirming the upper critical dimension and characterizing phase transition behavior.

Contribution

It provides the first combined analytical and numerical analysis of the model, confirming the upper critical dimension and elucidating phase transition properties.

Findings

Phase transition location determined

Critical exponents identified

No evidence of a glass phase found

Abstract

We investigate both analytically and numerically the ensemble of minimum-weight loops and paths in the negative-weight percolation model on random graphs with fixed connectivity and bimodal weight distribution. This allows us to study the mean-field behavior of this model. The analytical study is based on a conjectured equivalence with the problem of self-avoiding walks in a random medium. The numerical study is based on a mapping to a standard minimum-weight matching problem for which fast algorithms exist. Both approaches yield results which are in agreement, on the location of the phase transition, on the value of critical exponents, and on the absence of any sizeable indications of a glass phase. By these results, the previously conjectured upper critical dimension of d_u=6 is confirmed.