Permutation-matrix analogue of joint random–deterministic strong convergence

Develop a counterpart of joint strong convergence for families mixing random and deterministic matrices in the setting of random permutation matrices; specifically, prove strong convergence for polynomials P(U_1^N,…,U_r^N,B_1^N,…,B_s^N) with permutation U_i^N and strongly convergent deterministic B_j^N, paralleling existing results for GUE and Haar unitary models.

Background

Joint strong convergence for random and deterministic matrices is well developed in the GUE/Haar frameworks, relying on analytic tools such as Schwinger–Dyson equations and interpolation. However, analogous results for permutation matrices are not known.

Extending these techniques to discrete ensembles would broaden applicability to numerous combinatorial and geometric models where permutation matrices arise naturally.