Phase transitions in diluted negative-weight percolation models

TL;DR

This study explores how dilution affects the phase transition of negative-weight percolation on 2D lattices, revealing that certain types of dilution preserve the universality class while others alter it, through numerical and finite-size scaling analyses.

Contribution

It provides a detailed numerical analysis of how two different dilution methods impact the universality class of negative-weight percolation transitions.

Findings

Zero-weight dilution does not change the universality class.

Edge absence dilution leads to a different universality class.

Finite-size scaling was used to determine phase boundaries.

Abstract

We investigate the geometric properties of loops on two-dimensional lattice graphs, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of spanning loops of total negative weight. The resulting percolation problem is fundamentally different from conventional percolation, as we have seen in a previous study of this model for the undiluted case. Here, we investigate how the percolation transition is affected by additional dilution. We consider two types of dilution: either a certain fraction of edges exhibit zero weight, or a fraction of edges is even absent. We study these systems numerically using exact combinatorial optimization techniques based on suitable transformations of the graphs and applying matching algorithms. We perform a finite-size scaling analysis to obtain the phase diagram and determine the critical properties of the phase boundary. We find that the first type of dilution does not change the universality class compared to the undiluted case whereas the second type of dilution leads to a change of the universality class.