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The spectrum of random matrices has some universality properties, in the sense that they do not depend on details of the distribution of the matrix elements. For instance, the spectral gap, i.e. the smalles eigenvalue of your $AA^T$, should be the same as for real Gaussian $A$ matrices (when $A$ is real Gaussian, the ensemble of $AA^T$ is called the real Wishart ensemble). At least for $p=1/2$, this is actually proved in the paper Random Matrices: The Distribution of the Smallest Singular Values, by Terence Tao and Van Vu (Geom. Funct. Anal. Vol. 20, 260–297, 2010). For large dimensions, the eigenvalue density is called the Marchenko–Pastur distribution.