## Publications of the Department

Giardina', Cristian; Giberti, Claudio; Magnanini, Elena, (2021) -

*Approximating the Cumulant Generating Function of Triangles in the Erdös–Rényi Random Graph*- JOURNAL OF STATISTICAL PHYSICS, Articolo su rivista - Articolo in rivista (262) (, , ) - pagg. 1 - 22

**Abstract**: We study the pressure of the “edge-triangle model”, which is equivalent to the cumulant generating function of triangles in the Erdös–Rényi random graph. The investigation involves a population dynamics method on finite graphs of increasing volume, as well as a discretization of the graphon variational problem arising in the infinite volume limit. As a result, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.