Unification of the Nature’s Complexities via a Matrix Permanent—Critical Phenomena, Fractals, Quantum Computing, ♯P-Complexity

Abstract

We reveal the analytic relations between a matrix permanent and major nature’s complexities manifested in critical phenomena, fractal structures and chaos, quantum information processes in many-body physics, number-theoretic complexity in mathematics, and ♯P-complete problems in the theory of computational complexity. They follow from a reduction of the Ising model of critical phenomena to the permanent and four integral representations of the permanent based on (i) the fractal Weierstrass-like functions, (ii) polynomials of complex variables, (iii) Laplace integral, and (iv) MacMahon master theorem.

Keywords

• ♯p-complexity
• np-complexity
• critical phenomena
• fractals
• quantum computing
• matrix permanent
• macmahon master theorem
• toeplitz determinant