Local persistence in directed percolation

TL;DR

This paper improves estimates of the local persistence exponent in directed percolation across multiple dimensions, revealing non-trivial dependence on dimension and proposing a new related critical exponent based on advanced simulations.

Contribution

It introduces new algorithms and improved implementations for estimating persistence exponents in directed percolation, and explores their dimension dependence and related critical exponents.

Findings

Persistence exponent varies non-trivially with dimension for d > 4

Scaling corrections are significant in 2+1 and 3+1 dimensions

A new critical exponent ζ is proposed, matching η+δ in standard DP

Abstract

We reconsider the problem of local persistence in directed site percolation. We present improved estimates of the persistence exponent in all dimensions from 1+1 to 7+1, obtained by new algorithms and by improved implementations of existing ones. We verify the strong corrections to scaling for 2+1 and 3+1 dimensions found in previous analyses, but we show that scaling is much better satisfied for very large and very small dimensions. For d > 4 (d is the spatial dimension), the persistence exponent depends non-trivially on d, in qualitative agreement with the non-universal values calculated recently by Fuchs {\it et al.} (J. Stat. Mech.: Theor. Exp. P04015 (2008)). These results are mainly based on efficient simulations of clusters evolving under the time reversed dynamics with a permanently active site and a particular survival condition discussed in Fuchs {\it et al.}. These simulations suggest also a new critical exponent $\zeta$ which describes the growth of these clusters conditioned on survival, and which turns out to be the same as the exponent, \eta+\delta in standard notation, of surviving clusters under the standard DP evolution.