Spatio-temporal Correlations in the Manna Model in one, three and five dimensions

Gary Willis, Gunnar Pruessner

Published 2016-08-02Version 1

Although the paradigm of criticality is centred around spatial correlations and their anomalous scaling, very few studies of Self-Organised Criticality (SOC) are concerned with spatial correlations. Instead, integrated observables, such as avalanche size and duration, are commonly 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 SOC, measured numerically in detail in the Manna Model in d=1,3 and 5 dimensions and compared to theoretical results. Contrary to the notion held by some of SOC models organising into a critical state by re-arranging their spatial structure avalanche by avalanche, which may be expected to result in non-trivial spatial correlations in the quiescent state (between avalanches), correlations of inactive particles in the quiescent state are apparently short-ranged. Self-organisation, however, does take place as the (one-point) density of inactive particle organises into a particular profile that is independent of the driving location, also demonstrated analytically in one dimension. Activity and its correlations, on the other hand, display non-trivial long-ranged spatio-temporal scaling with exponents that can be related to established results, in particular avalanche size and duration exponents. The correlation length is 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 is recovered, with some corrections, in five dimensions.