Upper critical dimension of the negative-weight percolation problem

Abstract

By means of numerical simulations, we investigate the geometric properties of loops on hypercubic lattice graphs in dimensions d=2 through 7, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of system-spanning loops of total negative weight. The resulting negative-weight percolation (NWP) problem is fundamentally different from conventional percolation, as we have seen in previous studies of this model for the two-dimensional case. Here, we characterize the transition for hypercubic systems, where the aim of the present study is to get a grip on the upper critical dimension d u of the NWP problem. For the numerical simulations, we employ a mapping of the NWP model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms. We characterize the loops via observables similar to those in percolation theory and perform finite-size scaling analyses, e.g., three-dimensional hypercubic systems with side length up to L=56 sites, in order to estimate the critical properties of the NWP phenomenon. We find our numerical results consistent with an upper critical dimension d u=6 for the NWP problem.

Percolation thresholds on planar Euclidean relative-neighborhood graphs

“…I), p c = 0.798(3) (average size of the finite clusters). Further, the critical exponents ν = 1.33 (6) and β = 0.14(3) obtained from the order parameter and γ = 2.39(7), obtained from the scaling behavior of the average size of the finite clusters, are listed in Tab. I.…”

“…describing vortices in high T c superconductivity [3,4], the negativeweight percolation problem [5,6], and domain wall excitations in disordered media such as 2D spin glasses [7,8] and the 2D solid-on-solid model [9]. Besides discrete lattice models there is also interest in studying continuum percolation models, where recent studies reported on highly precise estimates of critical properties for spatially extended, randomly oriented and possibly overlapping objects with various shapes [10].…”

Monodominance in tropical forests: modelling reveals emerging clusters and phase transitions

“…Near phase transitions, the relationship between model parameters and most measurable quantities such as cluster size or correlation length (see the electronic supplementary material) can be described by power laws. The exponents of these power laws characterize phase transitions and are called critical exponents [39,40]. By checking the transitions in our model for these characteristic exponents, we could possibly link our specific model to the large class of spatial models and systems that show phase transitions and in which the geometry of interactions plays a decisive role.…”

Directed negative-weight percolation

“…Note that for standard directed percolation near a wall, for the results obtained using a series expansion the scaling relation is clearly violated [26]. Nevertheless, in the case of a violation it would be different from the undirected NWP case, where the standard scaling relations for percolation hold [6,8].…”