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arXiv · 1205.1412

Is negative-weight percolation compatible with SLE?

C. Norrenbrock·O. Melchert·A. K. Hartmann
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Abstract

We study numerically the geometrical properties of minimally weighted paths that appear in the negative-weight percolation (NWP) 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, the 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 (GS). A ground state on such a lattice can be determined performing a non-trivial transformation of the original graph and applying sophisticated matching algorithms. Here we examine whether the geometrical properties of the paths are in accordance with predictions of Schramm-Loewner evolution (SLE). Measuring the fractal dimension and reviewing Schramm's left passage formula indicates that the paths cannot be described in terms of SLE.

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C. Norrenbrock, O. Melchert, A. K. Hartmann. 2012-05-07. Is negative-weight percolation compatible with SLE?. https://doi.org/10.1103/physreve.87.032142

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