Analysis of the loop length distribution for the negative weight percolation problem in dimensions d=2 through 6

TL;DR

This study investigates the distribution of small negative-weight loops in the negative weight percolation problem across dimensions 2 to 6, revealing critical exponents and behaviors at the phase transition using exact combinatorial optimization algorithms.

Contribution

It provides the first detailed characterization of small loop distributions at criticality in NWP, connecting these to system-spanning loop phenomena through numerical analysis.

Findings

Identified critical exponents for small loop distributions.

Established relation between small and spanning loop behaviors.

Performed large-scale exact numerical simulations across multiple dimensions.

Abstract

We consider the negative weight percolation (NWP) problem on hypercubic lattice graphs with fully periodic boundary conditions in all relevant dimensions from d=2 to the upper critical dimension d=6. The problem exhibits edge weights drawn from disorder distributions that allow for weights of either sign. We are interested in in the full ensemble of loops with negative weight, i.e. non-trivial (system spanning) loops as well as topologically trivial ("small") loops. The NWP phenomenon refers to the disorder driven proliferation of system spanning loops of total negative weight. While previous studies where focused on the latter loops, we here put under scrutiny the ensemble of small loops. Our aim is to characterize -using this extensive and exhaustive numerical study- the loop length distribution of the small loops right at and below the critical point of the hypercubic setups by means of two independent critical exponents. These can further be related to the results of previous finite-size scaling analyses carried out for the system spanning loops. For the numerical simulations we employed a mapping of the NWP model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms. This allowed us to study here numerically exact very large systems with high statistics.