Disordered Systems, Spanning Trees and SLE

TL;DR

This paper introduces a path minimization problem on planar graphs linked to critical percolation exploration paths, demonstrating its connection to SLE_6 and spanning trees through numerical analysis and boundary condition variations.

Contribution

It formulates a new path minimization model on planar graphs that captures SLE_6 behavior and relates to spanning trees, with numerical validation across different lattices.

Findings

Model reproduces SLE_6 scaling limit

Numerical tests confirm SLE properties on various lattices

Defines a growth process for trees with SLE as a boundary condition

Abstract

We define a minimization problem for paths on planar graphs that, on the honeycomb lattice, is equivalent to the exploration path of the critical site percolation and than has the same scaling limit of SLE_6. We numerically study this model (testing several SLE properties on other lattices and with different boundary conditions) and state it in terms of spanning trees. This statement of the problem allows the definition of a random growth process for trees on two dimensional graphs such that SLE is recovered as a special choice of boundary conditions.