Universality of the least singular value and singular vector delocalisation for L\'evy non-symmetric random matrices

In this paper we consider $N \times N $ matrices $D_{N}$ with i.i.d. entries all following an $a-$stable law divided by $N^{1/a}$. We prove that the least singular value of $D_{N}$, multiplied by $N$, tends to the same law as in the Gaussian case, for almost all $a \in (0,2)$. This is proven by considering the symmetrization of the matrix $D_{N}$ and using a version of the three step strategy, a well known strategy in the random matrix theory literature. In order to apply the three step strategy, we also prove an isotropic local law for the symmetrization of matrices after slightly perturbing them by a Gaussian matrix with a similar structure. The isotropic local law is proven for a general class of matrices that satisfy some regularity assumption. We also prove the complete delocalization for the left and right singular vectors of $D_{N}$ at small energy, i.e., for energies at a small interval around $0$.