On the Unification of Nature’s Complexities via a Matrix Permanent: Critical Phenomena, Fractals, Quantum Computing, #P/NP-Complexity

We find a remarkable explicit connection between the major types of complexity in nature [Entropy 22, 322 (2020)]. They represent the critical phenomena, fractal structures in the theory of chaos, quantum information processing in many-body physics, cryptography, number-theoretic complexity in mathematics, and #P-complete problems. We show that all of them are analytically related to a well-known in mathematics matrix permanent via the fractal Weierstrass-like functions and polynomials or determinants involving complex variables. We discuss the #P/NP-complexity of quantum computing, nontrivial reduction of the critical phenomena problem to a permanent, new integral representations of the permanent revealing its relation to fractals and chaos, complex stochastic multivariate polynomials, number-theoretical functions, asymptotics of a Toeplitz determinant employed in the Onsager’s solution of the Ising model and given by the Szego limit theorems.