Spectral gap of convex combination of a random permutation and a bistochastic matrix

We consider a random bistochastic matrix of size N of the form (1-r)M + rQ where 0 < r < 1, M is a uniformly distributed permutation and Q is a given bistochastic matrix. Under sparsity and regularity assumptions on the *-distribution of Q, we prove that the second largest eigenvalue (1-r)M + rQ is essentially bounded by an approximation of the spectral radius of a deterministic asymptotic equivalent given by free probability theory.