Average properties of combinatorial problems and thermodynamics of spin models on graphs

The study of thermodynamic properties of classical spin models on infinite graphs naturally leads to consider the new combinatorial problems of random-walks and percolation on the average. Indeed, spin models with O(n) continuous symmetry present spontaneous magnetization only on transient on the average graphs, while models with discrete symmetry (Ising and Potts) are spontaneously magnetized on graphs exhibiting percolation on the average. In this paper we define the combinatorial problems on the average, showing that they give rise to classifications of graph topology which are different from the ones obtained in usual (local) random-walks and percolation. Furthermore, we illustrate the theorem proving the correspondence between Potts model and average percolation.