Surface Shape and Local Critical Behaviour in Two-Dimensional Directed Percolation

TL;DR

This paper investigates how a free surface influences the critical behavior of two-dimensional directed percolation, revealing surface relevance criteria, calculating correlation functions, and analyzing cluster properties through simulations and mean-field theory.

Contribution

It introduces a scaling framework for surface effects in directed percolation, computes correlation functions, and explores the impact of surface perturbations on critical cluster properties.

Findings

Surface is relevant when k<1/z, affecting local critical behavior.

Tip-to-bulk correlation function calculated in mean-field approximation.

Monte Carlo simulations reveal nonuniversal scaling and stretched exponential behavior.

Abstract

Two-dimensional directed site percolation is studied in systems directed along the x-axis and limited by a free surface at y=\pm Cx^k. Scaling considerations show that the surface is a relevant perturbation to the local critical behaviour when k<1/z where z=\nu_\parallel/\nu is the dynamical exponent. The tip-to-bulk order parameter correlation function is calculated in the mean-field approximation. The tip percolation probability and the fractal dimensions of critical clusters are obtained through Monte-Carlo simulations. The tip order parameter has a nonuniversal, C-dependent, scaling dimension in the marginal case, k=1/z, and displays a stretched exponential behaviour when the perturbation is relevant. The k-dependence of the fractal dimensions in the relevant case is in agreement with the results of a blob picture approach.