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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.
Highlights
We find a remarkable explicit connection between the major types of complexity in nature.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 in the theory of computational complexity
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
The analysis is based on the concept of the ]P/NP-complexity of computations and quantum information processing and computing (Section 2) as well as on a nontrivial reduction of the critical phenomena problem to a permanent (Section 3) and new integral representations of the permanent revealing its deep explicit relation to the fractals and chaos (Section 4), complex stochastic multivariate polynomials (Section 5), number-theoretical functions (Section 6), asymptotics of a Toeplitz determinant employed in the Onsager’s solution of Ising model and given by the Szegő limit theorems (Section 7)
Summary
We find a remarkable explicit connection between the major types of complexity in nature 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 in the theory of computational complexity. The analysis is based on the concept of the ]P/NP-complexity of computations and quantum information processing and computing (Section 2) as well as on a nontrivial reduction of the critical phenomena problem to a permanent (Section 3) and new integral representations of the permanent revealing its deep explicit relation to the fractals and chaos (Section 4), complex stochastic multivariate polynomials (Section 5), number-theoretical functions (Section 6), asymptotics of a Toeplitz determinant employed in the Onsager’s solution of Ising model and given by the Szegő limit theorems (Section 7).
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