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

Abstract

Let $z\in \mathbb {C}$, let $\sigma ^2>0$ be a variance, and for $N\in \mathbb {N}$ define the integrals

$$\begin{aligned} E_N^{}(z;\sigma ) := \left\{ \begin{array}{ll} {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}}\! (x^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x^2}}{\sqrt{2\pi }}dx&{}\quad \text{ if }\, N=1,\\ {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}^N}\! \prod \prod \limits _{1\le k<l\le N}\!\! e^{-\frac{1}{2N}(1-\sigma ^{-2}) (x_k-x_l)^2} \prod _{1\le n\le N}\!\!\!\!(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x_n^2}}{\sqrt{2\pi }}dx_n &{}\quad \text{ if }\, N>1. \end{array}\right. \!\!\! \end{aligned}$$

These are expected values of the polynomials $P_N^{}(z)=\prod _{1\le n\le N}(X_n^2+z^2)$ whose 2N zeros $\{\pm i X_k\}^{}_{k=1,\ldots ,N}$ 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.

Appendix A: Alternate Proof of Corollary 2.3

Proof of Corollary 2.3

By the permutation symmetry in $\{x_1,\ldots ,x_N\}$ of the pdf given by the integrand of (37), for the pertinent random variables $\{X_1,\ldots ,X_N\}$ we have

$$\begin{aligned} N\mathrm{{Var}}_N\!\left[ X_1\right] = {\textstyle {\sum \limits _{k=1}^N}}\mathrm{{Var}}_N\!\left[ X_k\right] , \end{aligned}$$

(135)

and for these (as for any) mean-zero random variables $\{X_1,\ldots ,X_N\}$, we have

$$\begin{aligned} {\textstyle {\sum \limits _{k=1}^N}}\mathrm{{Var}}_N\!\left[ X_k\right] = \mathrm{{Exp}}_N\left[ {\textstyle {\sum \limits _{k=1}^N}} X_k^2\right] . \end{aligned}$$

(136)

Under a Euclidean transformation (here: rotation) from $\{x_1,\ldots ,x_N\}$ to $\{y_1,\ldots ,y_N\}$,

$$\begin{aligned} \mathrm{{Exp}}_N\left[ {\textstyle {\sum \limits _{k=1}^N}} X_k^2\right] = \mathrm{{Exp}}_N\left[ {\textstyle {\sum \limits _{k=1}^N}} Y_k^2\right] . \end{aligned}$$

(137)

By the proof of Proposition 2.1,

$$\begin{aligned} \mathrm{{Exp}}_N\left[ {\textstyle {\sum \limits _{k=1}^N}} Y_k^2\right] = \mathrm{{Exp}}_N\left[ Y_1^2\right] + (N-1)\mathrm{{Exp}}_N\left[ Y_2^2\right] = \sigma ^2 + N-1. \end{aligned}$$

(138)

The proof is complete. $\square$

Remark 8.1

There is no unique rotation in $\mathbb {R}^N$ which maps $\{x_1,x_2,\ldots ,x_N\}$ into $y_1:={\textstyle {\frac{1}{\sqrt{N}}}}\sum _{1\le k \le N} x_k$; even after stipulating that the $x_1$ axis should be rotated into the $y_1$ axis we are left to choose an arbitrary $SO(N-1)$ rotation “about the $y_1$-axis” to fix the remaining $N-1$ axes. Such a freedom may be useful to simplify the expected value integrals in the y coordinates, but we have not explored this here.

1.1 Appendix B: Alternate Proof of the Positivity of $E_{2K}^{}(z;\sigma );\; K\in \mathbb {N}$, $\sigma \ge 1$

Proposition 8.2

Let $K\in \mathbb {N}$. Then for $\sigma \ge 1$ and $z^2\in \mathbb {R}$ we have $E_{2K}(z;\sigma )\ge 0$, with “$=0$” iff $\sigma =1$ and $z^2=-1$.

Proof of Proposition 8.2

We split the $N=2K$ variables into two disjoint sets of size K, keeping the notation $x_n^{}$ for $n=1,\ldots ,K$ and renaming $x_n^{}=:\tilde{x}_k^{}$ if $n=K+k$ with $k=1,\ldots ,K$. Note that

$$\begin{aligned} \sum \!\sum _{1\le k,l\le 2K} (x_k-x_l)^2 = \sum \!\sum _{1\le k,l\le K} (x_k-x_l)^2 + \sum \!\sum _{1\le k,l\le K} (\tilde{x}_k-\tilde{x}_l)^2 + 2 \sum \!\sum _{1\le k,l\le K} (x_k-\tilde{x}_l)^2,\qquad \end{aligned}$$

(139)

and further that

$$\begin{aligned} \sum \!\sum _{1\le k,l\le K} (x_k-\tilde{x}_l)^2 = K\!\!\sum _{1\le k\le K}\!\!x_k^2 + K\!\!\sum _{1\le l\le K}\!\!\tilde{x}_l^2 - 2\sum \!\sum _{1\le k,l\le K}x_k\tilde{x}_l. \end{aligned}$$

(140)

Again recalling that $(x_1^{},\ldots ,x_K^{})=:\vec {x}$, and defining

$$\begin{aligned} \displaystyle \prod \!\!\prod _{1\le k<l\le K}\! e^{-\frac{1}{4K}(1-\frac{1}{\sigma ^2}) (x_k-x_l)^2}\!\!\! \prod _{1\le n\le K}\!\!(z^2+ x_n^2)e^{- \frac{1}{4}(1+\frac{1}{\sigma ^2}) x_n^2} =: \Upsilon (\vec {x}) \end{aligned}$$

(141)

and

$$\begin{aligned} \displaystyle \prod \!\!\prod _{1\le k,l\le K}\! e^{\frac{1}{2K}(1-\frac{1}{\sigma ^2}) x_k\tilde{x}_l} =: \Omega (\vec {x},\vec {\tilde{x}}), \end{aligned}$$

(142)

we have, for $K\in \mathbb {N}$,

$$\begin{aligned} \begin{array}{rl} E_{2K}^{}(z;\sigma )\! = &{} \!\! {\textstyle \frac{1}{\sigma {({2\pi }})^K}} \displaystyle \int _{\mathbb {R}^K}\!\! \int _{\mathbb {R}^K} \Upsilon (\vec {x}) \Omega (\vec {x},\vec {\tilde{x}}) \Upsilon (\vec {\tilde{x}}) d^{^K}\!\!x\,d^{^K}\!\!\tilde{x}\\ = &{} \!\! {\textstyle \frac{1}{\sigma {({2\pi }})^K}} \sum \limits _{j=0}^\infty \frac{1}{j!} \left( \frac{1}{2K}(1-\frac{1}{\sigma ^2})\right) ^j \displaystyle \int _{\mathbb {R}^K}\!\! \int _{\mathbb {R}^K} \Upsilon (\vec {x}) \Upsilon (\vec {\tilde{x}}) \Bigl (\,\textstyle \sum \!\sum \limits _{1\le k,l\le K}\!\!x_k\tilde{x}_l\Bigr )^j\!\! d^{^K}\!\!x\,d^{^K}\!\!\tilde{x}\!\!\!\!\!\\ = &{} \!\! {\textstyle \frac{1}{\sigma {({2\pi }})^K}} \sum \limits _{j=0}^\infty \frac{1}{j!} \left( \frac{1}{2K}(1-\frac{1}{\sigma ^2})\right) ^j \displaystyle \biggl (\int _{\mathbb {R}^K} \Upsilon (\vec {x}) \Bigl (\textstyle \sum \limits _{\; 1\le k\le K}\!\!x_k\Bigr )^j d^{^K}\!\!x\biggr )^2, \end{array}\qquad \end{aligned}$$

(143)

which, since $\sigma \ge 1$, is manifestly $\ge 0$. More precisely, r.h.s. (143) is estimated from below by the $j=0$ contribution, evaluating to $\frac{1}{\sigma }\Big [\sqrt{2\frac{\sigma ^2}{\sigma ^2+1} }(z^2 +2\frac{\sigma ^2}{\sigma ^2+1} )\Big ]^{2K}\ge 0$, with “$=0$” iff $\sigma =1$ and $z^2=-1$. When $\sigma =1$, the $j=0$ term is also the only contribution to r.h.s. (143). $\square$