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〈 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) 

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: 

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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, 

## III. AN EFFECTIVE FIELD THEORY 

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 

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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 

where the effective couplings J $^{(F)}$, J $^{(SG)}$, J (F) 0 and J (SG) 0 are given by 

For a constant short-range coupling distributed as in (11) 

and for the bimodal distribution (12), 

For the correlation functions we have C $^{(Σ)}$, Σ=F,SG, where 

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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 

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