On the permanents of circulant and degenerate Schur matrices

We communicate three formulas for the permanents of degenerate Schur and circulant matrices. These combinatorial and integral formulas are intended for the analytical and asymptotic evaluation of the permanents as well as for the solution of three-dimensional Ising model. The paper's goal is to draw attention to the open fundamental problem of finding the permanents' asymptotics. A solution to this problem would be tremendously important for physics of many-body systems and critical phenomena, as well as for quantum field theory.