Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-variate Normal Random Polynomials

• Published in: Journal of statistical physics 2017-10, Vol.169 (1), p.63-106
• Main Author: Kiessling, Michael Karl-Heinz
• Format: Article
• Language: English
• Subjects: Algebra; Asymptotic properties; Computer algebra; Curvature; Entropy; Integrals; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Polynomials; Proving; Quantum Physics; Random variables; Statistical Physics and Dynamical Systems; Theoretical; Variance
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-017-1843-6

Let z ∈ C , let σ 2 > 0 be a variance, and for N ∈ N define the integrals E N ( z ; σ ) : = 1 σ ∫ R ( x 2 + z 2 ) e - 1 2 σ 2 x 2 2 π d x if N = 1 , 1 σ ∫ R N ∏ ∏ 1 ≤ k < l ≤ N e - 1 2 N ( 1 - σ - 2 ) ( x k - x l ) 2 ∏ 1 ≤ n ≤ N ( x n 2 + z 2 ) e - 1 2 σ 2 x n 2 2 π d x n if N > 1 . These are expected values of the polynomials P N ( z ) = ∏ 1 ≤ n ≤ N ( X n 2 + z 2 ) whose 2 N zeros { ± i X k } k = 1 , … , N are generated by N identically distributed multi-variate mean-zero normal random variables { X k } k = 1 N with co-variance Cov N ( X k , X l ) = ( 1 + σ 2 - 1 N ) δ k , l + σ 2 - 1 N ( 1 - δ k , l ) . The E N ( z ; σ ) are polynomials in z 2 , explicitly computable for arbitrary N , yet a list of the first three E N ( z ; σ ) shows that the expressions become unwieldy already for moderate N —unless σ = 1 , in which case E N ( z ; 1 ) = ( 1 + z 2 ) N for all z ∈ C and N ∈ N . (Incidentally, commonly available computer algebra evaluates the integrals E N ( z ; σ ) only for N up to a dozen, due to memory constraints). Asymptotic evaluations are needed for the large- N regime. For general complex z these have traditionally been limited to analytic expansion techniques; several rigorous results are proved for complex z near 0. Yet if z ∈ R one can also compute this “infinite-degree” limit with the help of the familiar relative entropy principle for probability measures; a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to signed or complex measures governs the N → ∞ asymptotics of the regime i z ∈ R . Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.