Numerical Study of Local and Global Persistence in Directed Percolation

TL;DR

This study investigates local and global persistence probabilities in one-dimensional directed percolation, revealing universal algebraic decay exponents at criticality through Monte Carlo simulations, and establishing their independence from initial conditions.

Contribution

It provides the first detailed numerical analysis of both local and global persistence exponents in directed percolation, highlighting their universality and contrasting decay behaviors.

Findings

Both P_l(t) and P_g(t) decay algebraically with exponents ~1.50 at criticality.

The exponents are independent of initial density and microscopic dynamics.

P_l(t) relates to a return probability in directed percolation with an active source.

Abstract

The local persistence probability P_l(t) that a site never becomes active up to time t, and the global persistence probability P_g(t) that the deviation of the global density from its mean value rho(t)-<\rho(t)> does not change its sign up to time t are studied in a one-dimensional directed percolation process by Monte Carlo simulations. At criticality, starting from random initial conditions, both P_l(t) and P_g(t) decay algebraically with exponents theta_l ~ theta_g ~ 1.50(2), which is in contrast to previously known cases where theta_g < theta_l. The exponents are found to be independent of the initial density and the microscopic details of the dynamics, suggesting that theta_l and theta_g are universal exponents. It is shown that in the special case of directed-bond percolation, P_l(t) can be related to a certain return probability of a directed percolation process with an active source (wet wall).