Phase Diagram of Optimal Paths

TL;DR

This paper explores the phase diagram of optimal paths in disordered media, introducing an efficient algorithm and identifying various phases, including a new strong-disorder phase with self-affine paths, linking multiple models like percolation and polymers.

Contribution

It presents a novel algorithm for numerically solving the d-dimensional model and maps out the phase boundaries, revealing a new strong-disorder phase and unifying different disordered media problems.

Findings

Identified phase boundaries of the directed polymer universality class.

Discovered a new strong-disorder phase with self-affine paths.

Linked various models such as percolation and random walk within the phase diagram.

Abstract

We show that choosing appropriate distributions of the randomness, the search for optimal paths links diverse problems of disordered media like directed percolation, invasion percolation, directed and non-directed spanning polymers. We also introduce a simple and efficient algorithm, which solves the d-dimensional model numerically in order N^(1+d_f/d) steps where d_f is the fractal dimension of the path. Using extensive simulations in two dimensions we identify the phase boundaries of the directed polymer universality class. A new strong-disorder phase occurs where the optimum paths are self-affine with parameter-dependent scaling exponents. Furthermore, the phase diagram contains directed and non-directed percolation as well as the directed random walk models at specific points and lines.