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We take a first small step to extend the validity of Rudelson-Vershynin typeestimates to some sparse random matrices, here random permutation matrices. Wegive lower (and upper) bounds on the smallest singular value of a large randommatrix D+M where M is a random permutation matrix, sampled uniformly, and D isdiagonal. When D is itself random with i.i.d terms on the diagonal, we obtain aRudelson-Vershynin type estimate, using the classical theory of random walkswith negative drift.

Smallest Singular Value for Perturbations of Random Permutation Matrices