In this work we discuss a short range version of the p-spin model. The model is provided with a parameter that allows to control the crossover with the mean field behaviour. We detect a discrepancy between the perturbative approach and numerical simulation. We attribute it to non-perturbative effects due to the finite probability that each particular realization of the disorder allows for the formation of regions where the system is less frustrated and locally freezes at a higher temperature.

We report the value of the dynamical critical exponent z for the six dimensional Ising spin glass, measured in three different ways: from the behavior of the energy and the susceptibility with the Monte Carlo time and by studying the overlap-overlap correlation function as a function of the space and time. All three results are in a very good agreement with the Mean Field prediction z = 4. Finally we have studied numerically the remanent magnetization in 6 and 8 dimensions and we have compared it with the behavior observed in the SK model, that we have computed analytically.

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We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for T→Tc+. First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order O(1/N) and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian L for the fields δQαβi(t1, t2) up to the cubic order in the series expansion around the dynamic Mean-Field value $\overline{Q^{\alpha \beta}}(t_{1}, t_{2})$. We find the more general expression for the time depending non-local fluctuations, the propagators [〈δQαβi(t1, t2)δQαβj(t3, t4)〉ξ]J, in the quadratic approximation. Finally we compare the long-range limit of the correlations, derived in this formalism, with the correlations of the infinite-range model studied with the previous approach (random matrices)

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