AFreeenergydensityf computation

The detailed derivation of the average free energy densityf = 1Nβ [logZ]DM in (9) using the replicamethodisillustrated. In the last line, the asymptotic independence betweenhaw and (s0,sΨ) are applied as discussedin[7]. Theoverall construction of generations is graphically represented in Figure 1. Note that the solution of the first equation in (70) automatically satisfies the second equation (sub-gradient condition) since|tanh(K0)| 1,which indicates that J1 =J,Ja =0,a 2isonevalidsolution. Toachieve this, we compare certain properties of the orthogonal matrix generated from the diagonalization of the covariance matrixC with the orthogonal matrix which isactually generated from the Haar orthogonal measure. Specifically, we compute the cumulants of the trace of the powerk of the orthogonal matrix.