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.

Exact Recursive Calculation of Circulant Permanents: A Band of Different Diagonals inside a Uniform Matrix

“…The permanents of the order 6 and lower in the right-hand side of the recurrence equations (that is, A 3 , A( * ) 3 , B 4 , B 5 , etc.) should be computed directly via the definition of the permanent in Equation (1). Direct numerical calculations, such as in Figure 1, confirm that the recurrence ( 44)- (48) gives easy and fast access to the correct result for the permanent of the circulant (3) with arbitrary, even complex values of the entries c 0 , c 1 , c 2 for any, arbitrarily large matrix size n in linear time.…”

“…Plugging the third equation into Equation (30), we get the first equation of the system (27). At last, the second equation of the system (27) follows from the first equation and the representation of the permanent of the matrix A (1) with a 1's-column defect at the position of the first column obtained by means of Lemma 1:…”

“…Overall, this system of linear homogeneous recurrence relations with variable coefficients is of the 24-th order. One can go further on, and similarly, exclude the variable C (1) n by solving Equation (46) for C…”

“…This fact predetermines their fundamental role in the quantum theory of many-body systems, which are either bosonic (symmetric) or fermionic (anti-symmetric). The permanents are well-known in mathematical physics, especially in quantum computing science and the quantum field theory of interacting Bose fields [1][2][3][4][5][6][7][8]. However, compared to the determinants, the permanents are much more difficult to compute and they account for much more complicated many-body phenomena, such as the critical phenomena in phase transitions.…”

“…Importantly, the permanents play a significant role in the algebraic complexity theory as universal polynomials [28]. Nevertheless, despite amazing recent activity and interest (see, e.g., [1,[5][6][7][8]15,18,22,[29][30][31][32][33][34][35][36][37][38] and references therein), the methods for analyzing and calculating the permanents and their asymptotics for most of large matrices remain illusive.…”