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Conservation laws for strings in the Abelian Sandpile Model

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arxiv 1002.3974 v1 pith:IXIAFJEX submitted 2010-02-21 cond-mat.stat-mech hep-latmath.COnlin.CG

Sergio Caracciolo , Guglielmo Paoletti , Andrea Sportiello This is my paper · ORCID
classification cond-mat.stat-mechhep-latmath.COnlin.CG
keywords momentumperiodicstringsabeliancalllawspatternsrelation
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The Abelian Sandpile generates complex and beautiful patterns and seems to display allometry. On the plane, beyond patches, patterns periodic in both dimensions, we remark the presence of structures periodic in one dimension, that we call strings. We classify completely their constituents in terms of their principal periodic vector k, that we call momentum. We derive a simple relation between the momentum of a string and its density of particles, E, which is reminiscent of a dispersion relation, E=k^2. Strings interact: they can merge and split and within these processes momentum is conserved. We reveal the role of the modular group SL(2,Z) behind these laws.

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    math.AP 2026-07 accept novelty 7.0 of 10

    The minimal tropical roof through N marked points, scaled by 1/sqrt(N), converges uniformly to the zero-boundary Monge–Ampère solution for the limiting empirical measure.

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