Although the paradigm of criticality is centred around spatial correlations and their anomalous scaling, not many studies of Self-Organised 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 d = 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 organising into a critical state by re-arranging their spatial structure avalanche by avalanche, which may be expected to result in large, non-trivial, 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 increase with the system size, although they display (noisy) power law scaling over a range linear in the system size. Selforganisation, however, does take place as the (one-point) density of inactive particles organises 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 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 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 is recovered, with some corrections, in five dimensions.

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We introduce and discuss a real-space renormalization group (RSRG) procedure on very small lattices, which in principle does not require any of the usual approximations, e.g., a cut-off in the expansion of the Hamiltonian in powers of the field. The procedure is carried out numerically on very small lattices (4 × 4 to 2 × 2) and implemented for the Ising Model and the q = 3, 4, 5-state Potts Models. Nevertheless, the resulting estimates of the correlation length exponent and the magnetization exponent are typically within 3-7% of the exact values. The 4-state Potts Model generates a third magnetic exponent, which seems to be unknown in the literature. A number of questions about the meaning of certain exponents and the procedure itself arise from its use of symmetry principles and its application to the q 5 Potts Model. =