Stability of the replica symmetric solution in diluted perceptron learning

We study the role played by the dilution in the average behavior of a perceptron model with continuous coupling with the replica method. We analyze the stability of the replica symmetric solution as a function of the dilution field for the generalization and memorization problems. Thanks to a Gardner like stability analysis we show that at any fixed ratio $\alpha$ between the number of patterns M and the dimension N of the perceptron ($\alpha =M\mathrm{/}N$), there exists a critical dilution field $h_c$ above which the replica symmetric ansatz becomes unstable.