Course 2S620, Lecture 3 Sherrington-Kirkpatrick model

The mean-field theory of spin-glasses is a solvable model which shows the existence of a new kind of phase transition. The reference model is the Sherrington-Kirkpatrick model, introduced in 1975. After some false starts, a meaningfull solution was pro-posed by G. Parisi in the eighties using a very clever ansatz for the symmetry breaking. The results was obtained by using the non-rigorous method of replicas, which we are going to review later. In 2005 the joint effort of F. Guerra and M. Talagrand allowed to establish the correctness of Parisi result as a theorem. While this has been an impressive achievement (it took 30 years!), it must also be noted that some relevant properties of the quenched equilibrium state, like ultrametricity, are still the subject of intensive investigations. Similarly to the Curie-Weiss model, the Sherrington-Kirkpatrick model is obtained by considering an interaction between all possible couples of N ∈ N spins. With ref-erence to the setting of the previous lecture, we define the volume Λ = {1, 2,..., N}, with cardinality |Λ | = N and the sample space ΩN = {−1,+1}N with cardinality |ΩN | = 2N. Let {Jij}i,j∈Λ×Λ be a family of i.i.d. random variables. While the model can be defined for an arbitrary choice of the distribution of the random variable we will stick to the choice of a standard normal distribution, that is for any (i, j) ∈ Λ×Λ the Jij are i.i.d. Gaussian random variable with E(Jij) = 0 E(J2ij) = 1 (1.1) The Hamiltonian then reads