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

We study the spectral gap behavior of an operator obtained by summing a random permutation $M$ and a deterministic bistochastic matrix $Q$. We are interested in the asymptotic in terms of dimension. In the case where $(M,Q)$ are asymptotically free with amalgamation over the diagonal, we can compute limit operators $(u,q)$ which give the weak limit spectral distribution. Therefore we introduce free with amalgamation operators that are suitable for computing the spectral gap limit of our operator in high dimensions. We then approximate the spectral radius of the corresponding limit operator and finally give an upper bound for the spectral radius of the finite-dimensional operator. In particular, we show that if the deterministic matrix underlying graph is an expander, then the underlying graph associated to the sum with a random permutation is again an expander.