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

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.