〈 O 〉 c ; J the Boltzmann-average of the quenched system (note that 〈 O 〉 c ; J depends on the given realization of the J ’s and of c : 〈 O 〉 = 〈 O 〉 c ; J ; for shortness we will often omit to write these dependencies) function, quadratic or not, there are two natural quanti- ties indicated by C (F) and C $^{(SG)}$, and that in turn will be calculated in terms of m (F) and m $^{(SG)}$, respectively. To avoid confusion, it should be kept in mind that in our approach, for any observable O there are - in principle - always two solutions that we label as F and SG, but, for any temperature, only one of the two solutions is stable and useful in the thermodynamic limit. In the following, we will use the label 0 to specify that we are referring to the pure model with Hamiltonian (1). Let m$_{0}$ ( βJ$_{0}$, βh ) be the stable magnetization of the pure model with coupling J$_{0}$ and in the presence of a uni- form external field h at inverse temperature β . Then, the order parameters m $^{(Σ)}$, Σ=F,SG, satisfy the follow- ing self-consistent decoupled equations and d P ( { J$_{i,j}$ } ) and P ( c ) are two product measures given in terms of two normalized measures dµ ( J$_{i,j}$ ) ≥ 0 and p ( c$_{i,j}$ ) ≥ 0, respectively: where the effective couplings J $^{(F)}$, J $^{(SG)}$, J (F) 0 and J (SG) 0 are given by The variables c$_{i,j}$ ∈ { 0 , 1 } specify whether a “long-range” bond between the sites i and j is present ( c$_{i,j}$ = 1) or absent ( c$_{i,j}$ = 0), whereas the J$_{i,j}$ ’s are the random vari- ables of the given bond ( i, j ). For the c$_{i,j}$ ’s, we shall consider the following distribution where c > 0. This choice leads in the thermodynamic limit N → ∞ to a number of long range connections per site distributed according to a Poisson law with mean connectivity c . In this paper for the J$_{i,j}$ ’s we will assume the distribu- tion For the short-range nearest-neighbor coupling, J$_{0}$ we will consider the distribution, except in the last case studied, the modified Edwards- Anderson model, where we consider, Depending on the temperature T, and on the param- eters of the probability distributions, dµ and p ( c$_{i,j}$ ), the random model may stably stay either in the paramag- netic (P), in the ferromagnetic (F), or in the spin glass (SG) phase. In our approach for the F and SG phases there are two natural order parameters that will be indi- cated by m (F) and m $^{(SG)}$. Similarly, for any correlation where C$_{0}$ ( βJ$_{0}$, βh ) is the correlation function of the pure model. For the corrective O (1 /N ) term in Eq. (18) we remind the reader to Eq. (33) of [12]. Let us indicate by C (1) and C (2) the averages and the quadratic averages over the disorder of the correlation function of degree, say k . Then, C (1) and C $^{(2)}$, are related to C (F) and C $^{(SG)}$, as follows For the correlation functions we have C $^{(Σ)}$, Σ=F,SG, where and for the bimodal distribution (12), For a constant short-range coupling distributed as in (11) 〈 O 〉 def = ∑ { σ$_{i}$ $_{}}$O c ; J e − βH $^{c}$;J ( { σ$_{i}$ } ) Z c ; J , (6) d P ( { J$_{i,j}$ } ) def = ∏ ( i,j ) ,i<j dµ ( J$_{i,j}$ ) , ∫ dµ ( J$_{i,j}$ ) = 1 , (7) P ( c ) def = ∏ ( i,j ) ,i<j p ( c$_{i,j}$ ) , ∑ c$_{i,j}$ =0 , 1 p ( c$_{i,j}$ ) = 1 . (8) m (Σ) = m$_{0}$ ( βJ (Σ) 0 , βJ $^{(Σ)}$m(Σ) + βh ) , (13) βJ (F) = c ∫ dµ ( J$_{i,j}$ ) tanh( βJ$_{i,j}$ ) , (14) βJ (SG) = c ∫ dµ ( J$_{i,j}$ ) tanh $^{2}$( βJ$_{i,j}$ ) . (15) J (F) 0 = a βJ (SG) 0 = tanh − $^{1}$(tanh$^{2}$( βa )) . (16) J (F) 0 = 0 βJ SG 0 = tanh − $^{1}$(tanh$^{2}$( βa )) (17) p ( c$_{ij}$ ) = c N δ$_{c}$$_{ij}$$_{,}$$_{1}$ + ( 1 − c N ) δ$_{c}$$_{ij}$$_{,}$$_{0}$, (9) dµ ( J$_{i,j}$ ) dJ$_{i,j}$ = δ ( J$_{i,j}$ − J ) , (10) dµ ( J$_{0}$ ) dJ$_{0}$ = δ ( J$_{0}$ − a ) , (11) dµ$_{0}$ ( J$_{0}$ ) dJ$_{0}$ = 1 2 δ ( J$_{0}$ − a ) + 1 2 δ ( J$_{0}$ + a ) . (12) C (Σ) = C$_{0}$ ( βJ (Σ) 0 , βJ $^{(Σ)}$m(Σ) + βh ) + O ( 1 N ) , (18) C (1) = C $^{(F)}$, in F , (19) C (1) = 0 , k odd , in SG , (20) C (1) = C $^{(SG)}$, k even , in SG , (21) III. AN EFFECTIVE FIELD THEORY 3