Phase transitions in diluted negative-weight percolation models

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 exhibits 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 in the universality class.

Figure 1

Samples of $L=64$ loop configurations for the two different types of disorder investigated. Percolating loops appear as dashed lines and are marked by arrows. (a) $\rho =0.1$, $p_{\text{I}}=0.3940$, where $p_{\text{I}}$ is the fraction of edges with weight equal to zero. A loop can include those edges without increasing its configurational energy. Edges with zero weight that are included in loops are colored gray. (b) $\rho =1.0$, $p_{\text{II}}=0.4998$, where $p_{\text{II}}$ is the fraction of missing edges on the lattice. Those edges that comprise the cluster of connected edges on which the percolating loop is located appear as thin gray lines.

Figure 2

(Color online) Illustration of the algorithmic procedure. (a) Original lattice $G$ with edge weights, (b) auxiliary graph $G_A$ with proper weight assignment: black edges carry the same weight as the respective edge in the original graph and gray edges carry zero weight. (c) Minimum-weight perfect matching $M$: bold edges are matched and dashed edges are unmatched. (d) Loop configuration (bold edges) that corresponds to the MWPM depicted in (c).

Figure 3

(Color online) Critical lines in the disorder-dilution planes. [(a) and (b)] Phase diagram for type I and type II dilutions, respectively. (a) signifies a domain in the disorder-dilution plane without percolating loops and (b) labels a domain where percolating loops are present.

Figure 4

(Color online) Scaling analysis to probe points along the critical lines in the disorder-dilution planes. (a) For a disorder parameter fixed to $\rho =0.25$, the scaling analysis of the percolation probability $P_L$ yields a critical value $p_{\text{I}}^c=0.184\left(2\right)$ for the dilution above which percolating loops appear in the system. It further yields a critical exponent $\nu =1.52\left(11\right)$. (b) For a disorder parameter fixed to $\rho =1.0$, the scaling analysis of the normalized average box size $A_L$ yields a critical value $p_{\text{II}}^c=0.4998\left(5\right)$ for the dilution parameter and a critical exponent $\nu =1.33\left(5\right)$.

Figure 5

(Color online) FSS of the average normalized loop length $P_L^{\infty }$ that leads to the estimate of $\beta$ listed in Table . Here, the data collapse is best near the critical point while there are corrections to scaling off criticality.

Figure 6

(Color online) Analysis of the fractal dimension $d_f$, the roughness exponent $d_r$, and the exponent $\gamma /\nu$ that relates to the fluctuations of the loop lengths at $\rho =1.0$ and $p_{\text{II}}=0.4998\left(5\right)$. (a) Estimation of the above exponents by means of a pure fit to the power-law data. (b) Estimation of the asymptotic exponents by extrapolation using the effective (local) exponents of the data points shown in (a).

Figure 7

(Color online) Detailed analysis at $\rho =1.0$ and $p_{\text{II}}$. Rescaled distribution $D_L\left(\ell \right)$ of the length $\ell$ of the largest loop found for each realization of the disorder. The inset shows the distribution $n_s$ of all loop lengths $s$, excluding the spanning loops, at the critical point. Gray data points were omitted from the fit that led to the estimate of $\tau =2.54\left(5\right)$.