Optimal self-avoiding paths in dilute random medium

TL;DR

This paper investigates how dilution affects optimal self-avoiding paths in a random medium, revealing a transition to fractal paths at the percolation threshold and analyzing associated scaling behaviors.

Contribution

It introduces a new finite size scaling analysis and renormalization group calculations to understand the effects of dilution on optimal paths, especially near the percolation threshold.

Findings

Optimal paths remain unchanged below the percolation threshold.

Paths become fractal with dimension D_min at the threshold.

Energy fluctuations scale with path length as L^ω, with ω≈1.02.

Abstract

By a new type of finite size scaling analysis on the square lattice, and by renormalization group calculations on hierarchical lattices we investigate the effects of dilution on optimal undirected self-avoiding paths in a random environment. The behaviour of the optimal paths remains the same as for directed paths in undiluted medium, as long as forbidden bonds are not exceeding the percolation threshold. Thus, overhanging configurations do not alter the standard self-affine directed polymer scaling regime, even above the directed threshold, when they become unavoidable. When dilution reaches the undirected threshold, the optimal path becomes fractal, with fractal dimension equal to $D_{\rm min}$, the dimension of the minimal length path on percolation cluster backbone. In this regime the optimal path energy fluctuation, $\overline{\Delta E}$, can be ascribed entirely to minimal length fluctuations, and satisfies $\overline{\Delta E} \propto L^{\omega}$, with $\omega=1.02 \pm 0.06$ in $2d$, $L$ being the Euclidean distance. Hierarchical lattice calculations confirm that $\omega$ is also the exponent of the leading scaling correction to $\overline E \propto L^{D_{\rm min}}$. Upon approaching threshold, the probability, ${\cal R}$, that the optimal path does not stick entirely on the minimal length one, obeys ${\cal R} \sim \left( \Delta p\right) ^{\rho}$, with $\rho \sim 1.0 \pm 0.05$ on hierarchical lattices. Such behaviour could be characteristic of the crossover to fractal regime. Transfer matrix results on square lattice show that a similar full sticking does not occur for directed paths at the directed percolation threshold.