Frustrated spherical spin model and walk statistics on Bethe lattices

It is known that the free energy of the spherical model of Berlin and Kac [Phys. Rev. 86, 821 (1952)], with nearest-neighbor couplings on an Euclidean lattice, can be expressed in terms of generating functions of walks on the same lattice. We show that this relationship is valid on any discrete geometrical structure (graph) and can be exploited to solve the model on non-Euclidean lattices, such as a variety of infinite tree graph. We show next that this technique of solution can be employed even when interactions are extended beyond nearest neighbors and frustrated, provided one defines distance in terms of shortest path on the graph between two sites. Implications of this definition for Euclidean lattices is discussed. We then compute the phase diagram of the spherical model with next-nearest-neighbor competing interactions on the usual Bethe lattice, and the thermodynamical behavior of ferromagnet and antiferromagnet on Bethe cactuses, that is, trees whose branches are symplexes.