Percolation in random environment

TL;DR

This paper investigates bond percolation on a square lattice with correlated randomness, revealing a transition from non-universal behavior to an infinite randomness fixed point with exactly known critical exponents.

Contribution

It introduces a model of correlated bond percolation and identifies a transition to an infinite randomness fixed point with precise critical exponents.

Findings

Weak disorder leads to non-universal exponents.

Strong disorder results in an infinite randomness fixed point.

Critical exponents are exactly known at strong disorder.

Abstract

We consider bond percolation on the square lattice with perfectly correlated random probabilities. According to scaling considerations, mapping to a random walk problem and the results of Monte Carlo simulations the critical behavior of the system with varying degree of disorder is governed by new, random fixed points with anisotropic scaling properties. For weaker disorder both the magnetization and the anisotropy exponents are non-universal, whereas for strong enough disorder the system scales into an {\it infinite randomness fixed point} in which the critical exponents are exactly known.