Explicit near-fully X-Ramanujan graphs

Let p(Y1, ..., Yd, Z1, ..., Ze) be a self-adjoint noncommutative polynomial, with coefficients from C
^r×r
, in the indeterminates Y1,..., Yd (considered to be self-adjoint), the indeterminates Z1, ..., Ze, and their adjoints Z1*, ..., Ze*. Suppose Y1, ..., Yd are replaced by independent random n x n matching matrices, and Z1, ..., Ze are replaced by independent random n x n permutation matrices. Assuming for simplicity that p's coefficients are 0-1 matrices, the result can be thought of as a kind of random rn-vertex graph G. As n goes to infinity, there will be a natural limiting infinite graph X that covers any finite outcome for G. A recent landmark result of Bordenave and Collins shows that for any , with high probability the spectrum of a random G will be eps-close in Hausdorff distance to the spectrum of X (once the suitably defined “trivial” eigenvalues are excluded). We say that G is “eps-near fully X-Ramanujan”. Our work has two contributions: First we study and clarify the class of infinite graphs X that can arise in this way. Second, we derandomize the Bordenave-Collins result: for any X, we provide explicit, arbitrarily large graphs G that are covered by X and that have (nontrivial) spectrum at Hausdorff distance at most eps from that of X. This significantly generalizes the recent work of Mohanty et al., which provided explicit near-Ramanujan graphs for every degree d (meaning d-regular graphs with all nontrivial eigenvalues bounded in magnitude by 2sqrt(d-1) + eps). As an application of our main technical theorem, we are also able to determine the “eigenvalue relaxation value” for a wide class of average-case degree-2 constraint satisfaction problems.