SPHERICAL MODEL ON THE BETHE LATTICE VIA THE STATISTICS OF WALKS

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Abstract
We give in explicit form a very general relation between the solution of the spherical model on a graph and the statistics of walks on the same graph. Such formulation sheds light on the role of topology in the statistical behaviour of model systems, since a graph is meant as a geometrical entity with no regard to any embedding space. This includes, of course, euclidean and fractal lattices as in particular cases. In this work we also show how this relation can be exploited to actually compute the thermodynamics of the model on a class of infinite graphs by giving the exact solution on Bethe lattices, where no euclidean metrics can be defined.

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