This paper describes some of the analytic tools developed recently by Ghirl
anda and Guerra in the investigation of the distribution of overlaps in the
Sherrington-Kirkpatrick spin glass model and of Parisi's ultrametricity. I
n particular, we introduce to this task a simplified (but also generalized)
model on which the Gaussian analysis is made easier. Moments of the Hamilt
onian and derivatives of the free energy are expressed as polynomials of th
e overlaps. Under the essential tool of self-averaging, we describe with fu
ll rigour, various overlap identities and replica independence that actuall
y hold in a rather large generality. The results are presented in a languag
e accessible to probabilists and analysts..