Spatio-temporal correlations in the Manna model in one, three and five dimensions

Abstract

Although the paradigm of criticality is centered around spatial correlations and their anomalous scaling, not many studies of self-organized criticality (SOC) focus on spatial correlations. Often, integrated observables, such as avalanche size and duration, are used, not least as to avoid complications due to the unavoidable lack of translational invariance. The present work is a survey of spatio-temporal correlation functions in the Manna Model of SOC, measured numerically in detail in [Formula: see text] = 1,3 and 5 dimensions and compared to theoretical results, in particular relating them to “integrated” observables such as avalanche size and duration scaling, that measure them indirectly. Contrary to the notion held by some of SOC models organizing into a critical state by re-arranging their spatial structure avalanche by avalanche, which may be expected to result in large, nontrivial, system-spanning spatial correlations in the quiescent state (between avalanches), correlations of inactive particles in the quiescent state have a small amplitude that does not and cannot increase with the system size, although they display (noisy) power law scaling over a range linear in the system size. Self-organization, however, does take place as the (one-point) density of inactive particles organizes into a particular profile that is asymptotically independent of the driving location, also demonstrated analytically in one dimension. Activity and its correlations, on the other hand, display nontrivial long-ranged spatio-temporal scaling with exponents that can be related to established results, in particular avalanche size and duration exponents. The correlation length and amplitude are set by the system size (confirmed analytically for some observables), as expected in systems displaying finite size scaling. In one dimension, we find some surprising inconsistencies of the dynamical exponent. A (spatially extended) mean field theory (MFT) is recovered, with some corrections, in five dimensions.