Journal of Statistical Physics, Vol. 100, Nos. 5/6, 2000. p. 871-892.
This paper describes some of the analytic tools developed recently by Ghirlanda and Guerra in the investigation of the distribution of overlaps in the Sherrington-Kirkpatrick spin glass model and of Parisi's ultrametricity. In particular, we introduce to this task a simplified (but also generalized) model on which the Gaussian analysis is made easier. Moments of the Hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps. Under the essential tool of self-averaging, we describe with full rigour, various overlap identities and replica independence that actually hold in a rather large generality. The results are presented in a language accessible to probabilists and analysts.