Exact asymptotic value of C_k^∞ for perfect matchings

Determine the exact value of the asymptotic improvement factor C_k^∞ := liminf_{n→∞} min_G perm(G)/B_S(k,n) for k-regular bipartite graphs on 2n vertices for k ≥ 4, where B_S(k,n) = ((k-1)^{k-1}/k^{k-2})^n is Schrijver’s bound.

Background

Beyond showing a strict separation from Schrijver’s bound, a sharper problem is to compute the precise asymptotic constant C_k^∞ for each fixed degree k. The authors discuss spectral and zeta-function methods toward this goal but leave the exact determination open.

The question is motivated by evidence that extremal graphs likely have large girth (e.g., Ramanujan expanders), in which the Bethe approximation becomes exact in the infinite-tree limit, making the finite-size correction central to the constant.