Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model

Abstract

We prove that ifĤ _N is the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity$\bar q_N = N^{ - 1} \sum \left\langle {S_l } \right\rangle _H^2$ converges in the variance to a nonrandom limit asN→∞, then the mean free energy of the model converges to the expression obtained by SK. Since this expression is known not to be correct in the low-temperature region, our result implies the “non-self-averaging” of the order parameter of the SK model. This fact is an important ingredient of the Parisi theory, which is widely believed to be exact. We also prove that the variance of the free energy of the SK model converges to zero asN→∞, i.e., the free energy has the self-averaging property.