Dynamic fluctuations in a Short-Range Spin Glass model

We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for $T\rightarrow T_{c}^{+}$. 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 $\delta Q^{\alpha\beta}_{i}(t_{1},t_{2})$ 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 $[\langle\delta Q^{\alpha\beta}_{i}(t_{1},t_{2}) \delta Q^{\alpha\beta}_{j}(t_{3},t_{4})\rangle_{\xi}]_{J}$, 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).