Physical Review E 76, 041126 (2007) Finite-size scaling of directed percolation in the steady state Hans-Karl Janssen * Institut f¨ ur Theoretische Physik III, Heinrich-Heine-Universit¨at, 40225 D¨ usseldorf, Germany Sven L¨ ubeck † Fachbereich Physik, Universit¨at-Duisburg-Essen, Campus Duisburg, 47048 Duisburg, Germany and Institut f¨ ur Theoretische Physik (B), RWTH Aachen, 52056 Aachen, Germany Olaf Stenull ‡ Fachbereich Physik, Universit¨at-Duisburg-Essen, Campus Duisburg, 47048 Duisburg, Germany (Dated: Received 9 May, 2007, published 22 October 2007) 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 absorb- ing 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 homoge- neous 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. PACS numbers: 64.60.Ak, 05.70.Jk, 64.60.Ht I. INTRODUCTION Critical phenomena like second order phase transitions are characterized by singularities of various quantities at the transition point (e.g. the specific heat, susceptibil- ity, correlation length). These singularities are described by power-laws governed by critical exponents. Studying the phase transition of a given system, one usually tries to identify the set of critical exponents which in con- junction with certain universal scaling functions charac- terizes the present universality class. Powerful analyti- cal and numerical techniques have been developed to ac- complish this task. Analytical investigations of universal quantities allow to address infinite system sizes but they are usually feasible only if one uses involved approxima- tions such as the diagrammatic perturbation expansions of renormalized field theory. Using numerical techniques * Electronic address: janssen@thphy.uni-duesseldorf.de † Electronic address: sven@thp.uni-duisburg.de ‡ Electronic address: olaf.stenull@uni-duisburg-essen.de like Monte Carlo simulations or transfer matrices calcu- lations one can avoid such approximations, however, the data is limited to finite systems sizes. Therefore, finite- size scaling (FSS) is widely used to extrapolate to the behavior of infinite systems. In particular, FSS is an efficient method to determine critical exponents and cer- tain universal scaling functions, and therefore, it often allows to identify the universality class (see Refs. [1, 2] for reviews). According to the phenomenological FSS theory [3], finite system sizes L result in a rounding and shifting of the critical singularities. It is assumed that finite-size effects in isotropic systems are controlled suffi- ciently close to the critical point by the ratio L/ξ ∞ , where ξ ∞ is the spatial correlation length of the infinite system. Approaching the transition point, this correlation length diverges as ξ ∞ ∝ r -ν , where r ∝|τ - τ c | measures the deviation of a temperature-like control parameter τ from its critical point value τ c , and where ν is the critical expo- nent of ξ ∞ . Finite-size effects decrease with increasing L and are negligible for L  ξ ∞ , i.e., for L 1/ν r  1, in sys- tems with periodic boundary conditions, true short range interactions, and without Goldstone modes. Otherwise,