Method of the matrix permanent in the theory of critical phenomena: Asymptotics for the large-size power-law circulant matrices

A microscopic theory of phase transitions in a critical region suggested recently in [V.V. Kocharovsky et al., Physica Scripta 90, 108002 (2015); Entropy 22, 322 (2020)] reduces calculation of an order parameter and various correlation functions to computing the permanents of certain matrices given by the well-known mean-field equations. It is known that an exact computation of a permanent is a #P-problem that can’t be solved by a classical computer in a polynomial time. Hence, finding adequate approximations and asymptotics of the permanent is of great importance. In this talk, we present our recent results in this direction for the case of power-law circulant matrices, including comparison with the known results on a random-phase approximation, exponential-law circulant matrices, and McCullagh asymptotics for doubly-stochastic matrices with a moderate variation of entries.