Annealed spherical model on random graphs

While classical spin models, such as the Ising and Potts systems, have long been analysed on regular and deterministic lattices, only recently have they been systematically investigated on sparse, heterogeneous networks. Originally introduced by Berlin and Kac in 1952, the spherical model replaces discrete +1, -1 spins on a d-dimensional lattice with real‐valued variables subject to a spherical constraint. Thanks to its quadratic Hamiltonian, diagonalization reduces the partition function to a constrained Gaussian integral, yielding an exact second‐order phase transition for d > 2 and mean‐field critical exponents above the upper critical dimension d_c = 4. A generalization of the spherical model is its formulation on graphs, with particular emphasis on the complete graph, in which each spin interacts with every other spin at equal strength but not with itself. It is solvable and the exact solution is equal to the mean-field model with the same critical exponents and the same thermodynamics functions. At the end, we plan to investigate the spherical model on a random graphs. In particular, we focus on the annealed setup and study the phase transition on Erdős-Rényi graph: edges are independent Bernoulli variables with parameter p, and spins remain continuous under the spherical constraint. We investigate two distinct cases: the dense case, in which the parameter p is very close to 1, thereby approximating the complete graph, and the sparse case, in which the mean number of connections for each sites remains finite. The main goal is to identify the critical temperature and the critical exponents in each of two distinct cases.