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Heuristic relative entropy principles with complex measures: large-degree asymptotics of a family of multi-variate normal random polynomials. (English) Zbl 1379.82008

Summary: Let \(z\in \mathbb {C}\), let \(\sigma^2>0\) be a variance, and for \(N\in \mathbb {N}\) define the integrals \[ E_N(z;\sigma) := \begin{cases}\frac{1}{\sigma} \int_{\mathbb {R}}(x^2+z^2) \frac{e^{-\frac{1}{2\sigma^2} x^2}}{\sqrt{2\pi}}dx\quad &\text{if } N=1,\\ \frac{1}{\sigma} \int_{\mathbb {R}^N} \mathop{\prod\prod}\limits_{1\leq k<l\leq N} e^{-\frac{1}{2N}(1-\sigma^{-2}) (x_k-x_l)^2} \prod\limits_{1\leq n\leq N}(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma^2} x_n^2}}{\sqrt{2\pi}}dx_n & \text{if } N>1. \end{cases} \] These are expected values of the polynomials \(P_N(z)=\prod_{1\leq n\leq N}(X_n^2+z^2)\) whose \(2N\) zeros \(\{\pm i X_k\}_{k=1,\dots,N}\) are generated by \(N\) identically distributed multi-variate mean-zero normal random variables \(\{X_k\}^{N}_{k=1}\) with co-variance \(\mathrm{Cov}_N(X_k,X_l)=(1+\frac{\sigma^2-1}{N})\delta_{k,l}+\frac{\sigma^2-1}{N}(1-\delta_{k,l})\). The \(E_N(z;\sigma)\) are polynomials in \(z^2\), explicitly computable for arbitrary \(N\), yet a list of the first three \(E_N(z;\sigma)\) shows that the expressions become unwieldy already for moderate \(N\) – unless \(\sigma = 1\), in which case \(E_N(z;1) = (1+z^2)^N\) for all \(z\in \mathbb {C}\) and \(N\in \mathbb {N}\). (Incidentally, commonly available computer algebra evaluates the integrals \(E_N(z;\sigma)\) 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 \(z\in \mathbb {R}\) 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 \(N\rightarrow \infty \) asymptotics of the regime \(iz\in \mathbb {R}\). Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

MSC:

82B35 Irreversible thermodynamics, including Onsager-Machlup theory
68W30 Symbolic computation and algebraic computation
15B52 Random matrices (algebraic aspects)
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