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

TL;DR

This paper investigates the large-degree asymptotics of multivariate normal polynomials with complex measures, extending the relative entropy principle to signed and complex measures, and provides rigorous proofs and conjectures for these asymptotics.

Contribution

It introduces a rigorous proof of the large-N asymptotics using the relative entropy principle for complex measures and conjectures its generalization to signed measures, with potential applications.

Findings

Asymptotic behavior of expected polynomials is governed by a generalized relative entropy principle.

Rigorous proof provided for the large-N limit with complex measures.

Evidence supporting the conjecture of a generalized entropy principle for signed measures.

Abstract

We study expected values of the polynomials $P_N^{}(z)=\prod_{1\leq n\leq N}(X_n^2+z^2)$ whose $2N$ zeros $\{\pm i X_k\}^{}_{k=1,...,N}$ are generated by $N$ identically distributed multi-variate mean-zero normal random variables $\{X_k\}^{N}_{k=1}$ with co-variance ${\rm{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma^2-1}{N})\delta_{k,l}+\frac{\sigma^2-1}{N}(1-\delta_{k,l})$. In principle these can be evaluated in closed form for arbitrary $N$, yet commonly available computer algebra handles only $N$ up to a dozen (due to memory constraints). A list of the first three expected polynomials shows that the expressions become unwieldy already for moderate $N$. On the other hand, asymptotic evaluations of the large-$N$ regime for complex $z$ have traditionally been limited to analytic expansion techniques, several rigorous results are proved about this regime for complex $z$. Yet if $z$ is real one can also compute the large-$N$ asymptotics in the "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 and complex measures}* governs the $N\to\infty$ asymptotics of the regime of imaginary $z$. Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.