Paths in the minimally weighted path model are incompatible with Schramm-Loewner evolution

We study numerically the geometrical properties of minimally weighted paths that appear in the minimally weighted path (MWP) model on two-dimensional lattices assuming a combination of periodic and free boundary conditions (BCs). Each realization of the disorder consists of a random fraction $(1-\rho )$ of bonds with unit strength and a fraction $\rho$ of bond strengths drawn from a Gaussian distribution with zero mean and unit width. For each such sample, a path is forced to span the lattice along the direction with the free BCs. The path and a set of negatively weighted loops form a ground state. A ground state on such a lattice can be determined performing a nontrivial transformation of the original graph and applying sophisticated matching algorithms. Here we examine whether the geometrical properties of the paths are in accordance with the predictions of the Schramm-Loewner evolution (SLE). Measuring the fractal dimension, considering the winding angle statistics, and reviewing Schramm's left passage formula indicate that the paths cannot be described in terms of SLE.

Figure 1

Illustration of minimum-weight configurations consisting of loops (gray) and one path (black) in a regular 2D square lattice of side length $L=64$. The boundary conditions are periodic in horizontal (left-right) direction and free in vertical direction. The path is forced to connect the middle of the lower boundary to the upper boundary. The minimum-weight configurations minimize Eq. (3). For a small value of the disorder parameter $\rho =0.2$, the path crosses the lattice in a rather straight fashion since the path weight is mainly determined by its length. Increasing the order parameter to $\rho =0.3413$, the path length increases as well and the path takes advantage of more negative-weighted edges. If the disorder parameter is increased to $\rho =0.6$, the loops and the path will become more dense.

Figure 2

(a) Illustration of a minimum-weight configuration consisting of loops (gray) and one path (black) in the unit disk with a radius of 40. The terminal points of the path are fixed on $-1+0i$ and $1+0i$ to ensure that they will be mapped as required by Cayley's function. All points which Schramm's left passage formula is checked for are located on the vertical solid line (deep gray). (b) The same minimum-weight configuration as in (a) after the conformal mapping by Cayley's function has been performed, so the path connects the origin to infinity in the upper half plane. The dashed lines indicate that the lattice is not displayed completely. Due to the mapping, the original, vertical line containing all checkpoints becomes a semicircle.

Figure 3

Illustration of the algorithmic procedure: Illustrated edges that are not marked by a weight visually carry a zero weight actually [except in (d) and (e)]. (a) The initial, weighted lattice graph. (b) Two extra nodes (gray) are added to render a path possible. They are connected to the lattice boundaries as described in the text. (c) Each edge is replaced by two nodes (black boxes) and three edges joining the original nodes and the added nodes in a row. The initial weights are assigned to one of the added edges (bold edges) that join one of the original nodes and one extra node. (d) The original nodes are duplicated preserving their neighbors. Additionally, the duplicated nodes are interconnected by an added edge carrying a zero weight. (e) A MWPM is computed: Bold edges are matched and dashed ones are unmatched. (f) Reconstruction to the initial lattice taking the MWPM result into account. The thickly denoted path and loop represent the minimally weighted configuration.

Figure 4

Estimation of the fractal dimension $d_f$ by means of a simple fit to the power-law data. The data points were obtained by measuring the averaged path length $\langle \ell \rangle$ on a square lattice scale $4:1$. The inset also shows estimation of the fractal dimension obtained by measuring $\langle \ell \rangle$ on the unit disk (circles) and by considering paths on the upper half plane (boxes) that have been created on the unit disk first and mapped [with Eq. (4)] to the upper half plane subsequently. $R$ describes the number of nodes which discretize the radius of the unit disk. Note that the upper dataset was shifted vertically for a more clear presentation. As a matter of fact, the circles and boxes almost lie on top of each other.

Figure 5

(a) Illustration of a minimum-weight configuration consisting of loops (gray) and one path (black) in a lattice of size $257\times 64$. Due to the “Manhattan-structure” of the underlying lattice, the semicircle ($R=32$) assumes the shape of a triangle (light gray). The boundaries at the top and bottom border (dashed) are free and the ones left and right (solid) are periodic. For testing, we also considered the case where also the left and right boundaries exhibit free boundary conditions. The edge weights are taken from a “Gauss-like” distribution shown in Eq. (2) featuring $\rho =0.3413$. The path is forced to connect the horizontal boundaries as described in the text. The minimum-weight configuration has to minimize Eq. (3). (b) Sketch of a lattice that approximates the upper half plane to check SLPF predictions at some certain places located on the semicircle. The checkpoints marked by rhombuses lie on the left and the other ones (crosses) on the right of the path (black). For every realization of the disorder, the predictions of SLPF are examined independently for each checkpoint. $\varphi$ states the angle between the ordinate and considered checkpoints in the upper half plane.

Figure 6

Comparison between the measured probabilities $p\left(\varphi \right)$ and the predictions of SLPF $P_{\kappa }\left(\varphi \right)$. The left passage probability $p\left(\varphi \right)$ has been measured for various points on a semicircle ($R=128$). $\varphi$ denotes the angle that is illustrated in Fig. 5 (right). The computational simulations have been performed at the critical point in a lattice of size $1025\times 256$ using 51200 realizations of the disorder. The chosen diffusion constant ${\kappa }^{☆}=3.343$ provides the finest agreement between $P_{\kappa }\left(\varphi \right)$ and $p\left(\varphi \right)$ (cf. Fig. 7). The right bottom inset illustrates the deviation of the measured left-passage probability from the SLE predictions. For reason of clarity, merely a few calculated data points are illustrated. As a matter of fact, $p\left(\varphi \right)$ has been estimated for 257 different values of $\varphi$. The upper left inset displays the same deviation but for the case where the starting point of the path is no fixed, for the best-fitting value ${\kappa }^{\mathrm{free}}=2.89\left(15\right)$. The systematic significant deviations indicate that here the left-passage formula does not hold.

Figure 7

The cumulative squared deviation $f_{SC}\left(\kappa \right)$ of the measured probabilities $p\left(\varphi \right)$ from the corresponding values given by Schramm's left passage formula $P_{\kappa }\left(\varphi \right)$ as a function of the diffusion constant $\kappa$. The semicircle, which contains the checkpoints, features a radius $R=128$ in a lattice of size $1025\times 256$. The simulations ($51200$ realizations of the disorder) have been performed at the critical point. The squared deviation has been calculated for discrete values of $\kappa$ between $2.5$ and $4.3$ with step range $0.001$. For reasons of clarity only a few error bars are displayed. The cumulative squared deviation yields the finest agreement between $p\left(\varphi \right)$ and $P_{\kappa }\left(\varphi \right)$ at ${\kappa }^{☆}=3.343\pm 0.065$.

Figure 8

(a) The diffusion constant ${\kappa }^{☆}$ depending on the ratio between the lateral extensions of the lattice. The horizontal extension $L_{\mathrm{x}}$ is varied in length and the radius $R$ of the semicircle is chosen to be the half of the system size $L=L_{\mathrm{y}}$. The estimates of the diffusion constant have been determined at the critical point by means of $51200$ realizations of the disorder. For reasons of clarity, the error bars are not illustrated. Note that all estimates (except for the shortest horizontal extensions until 80) are compatible to each other within the error bars. (b) The diffusion constant ${\kappa }^{☆}$ depending on the system size $L$ that denotes the number of nodes in the vertical direction of the lattice. The horizontal extension is equivalent to $4L+1$. The radius of the semicircle is chosen to be the half of the system size in every cases. The estimates of the diffusion constant have been determined at the critical point by means of $51200$ realizations of the disorder.

Figure 9

(a) Probability mass function of the roughness of the paths in a lattice of size $1025\times 256$. (b) Variance of the winding angle over the vertical width of the system. The lines indicate the result of the fit to Eq. (7), yielding ${\kappa }^{\circ }=2.45\left(8\right)$. The error bars are smaller than the symbol size. Both the data in (a) and each data point in (b) were obtained by considering $50000$ different paths.