Finite-size scaling of directed percolation in the steady state

TL;DR

This paper investigates finite-size scaling in directed percolation, a key non-equilibrium phase transition, through analytical and numerical methods, revealing new universal ratios and confirming results with simulations.

Contribution

It provides the first analytical derivation of finite-size scaling forms for directed percolation's moments and introduces a new ratio as a universal signature of the DP class.

Findings

Finite-size scaling forms derived analytically for DP moments.

Introduction of a new universal ratio analogous to Binder cumulant.

Monte Carlo simulations confirm analytical predictions.

Abstract

Recently, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in non-equilibrium systems at the instance of directed percolation (DP), which has become the paradigm of non-equilibrium phase transitions into absorbing states, above, at and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d-dimensional hypercube of finite edge length L with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a new signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.