Configurational statistics of densely and fully packed loops in the negative-weight percolation model

TL;DR

This study uses numerical simulations to analyze the geometric properties of densely packed loops in the negative-weight percolation model across multiple dimensions, revealing a transition to random-walk behavior in higher dimensions.

Contribution

It introduces a computational approach to characterize loop configurations in NWP models and identifies a dimensional threshold for random-walk behavior.

Findings

Loops behave like uncorrelated random walks in 3D and higher.

Finite-size scaling estimates geometric exponents of loop configurations.

Transition from non-random to random-walk behavior observed at d=3.

Abstract

By means of numerical simulations we investigate the configurational properties of densely and fully packed configurations of loops in the negative-weight percolation (NWP) model. In the presented study we consider 2d square, 2d honeycomb, 3d simple cubic and 4d hypercubic lattice graphs, where edge weights are drawn from a Gaussian distribution. For a given realization of the disorder we then compute a configuration of loops, such that the configurational energy, given by the sum of all individual loop weights, is minimized. For this purpose, we employ a mapping of the NWP model to the "minimum-weight perfect matching problem" that can be solved exactly by using sophisticated polynomial-time matching algorithms. We characterize the loops via observables similar to those used in percolation studies and perform finite-size scaling analyses, up to side length L=256 in 2d, L=48 in 3d and L=20 in 4d (for which we study only some observables), in order to estimate geometric exponents that characterize the configurations of densely and fully packed loops. One major result is that the loops behave like uncorrelated random walks from dimension d=3 on, in contrast to the previously studied behavior at the percolation threshold, where random-walk behavior is obtained for d>=6.