Logarithmic energy for random matrices

It was a conjecture of Shannon, proved by Artstein, Ball, Barthe and Naor [[1]], that the entropy $\operatorname{Ent}(X):=-\int f_X\log f_X$ is increasing along the central limit theorem: if $X_1,X_2,\ldots$ are i.i.d., centered random variables in $\mathrm L^2(\mathbb P)$ with a Lebesgue density $f$, then $$\operatorname{Ent}\left(\frac{X_1+\cdots+X_n}{\sqrt n}\right)$$ increases as $n\to\infty$ towards the entropy $\operatorname{Ent}(G)$ of a centered, Gaussian random variable $G$ with variance $\sigma^2:=\int x^2\,f(x)\,\mathrm dx$. Moreover, this characterizes the limiting distribution $\mathcal N(0,\sigma^2)$ since the entropy of any other absolutely continuous distribution with mean $0$ and variance $\sigma^2$ is strictly less than $\operatorname{Ent}(G)$. Together with [Djalil Chafaï](https://djalil.chafai.net/) and [Pierre Youssef](https://wp.nyu.edu/pyoussef/), we recently made public the paper [Monotonicity of the logarithmic energy for random matrices](https://arxiv.org/abs/2212.06090), which initiates the investigation of whether a similar phenomenon arises in the classical limit theorems of random matrix theory, such as Wigner's semicircular theorem, Tao–Vu's circular theorem, or Marčenko–Pastur's theorem, i.e., whether there is a simple “entropy” functional, on the eigenvalue distribution $\mu_A:=\frac1n\sum_{k=1}^n\delta_{\lambda_k(A)}$ of random matrices $A$, which is monotonic along these theorems. Voiculescu's free entropy $$\Sigma(\mu):=\iint\log{\lvert x-y\rvert}\,\mu(\mathrm dx)\mu(\mathrm dy)$$ appears to be a good candidate, for at least two reasons. First, it is already involved in the rate function of a Sanov-type large deviation principle for the empirical spectral distribution of Gauss−Hermite ensembles [[2]]. Second, it is at the heart of standard variational characterizations [[3]] of the semi-circular law $\mu_{\mathsf{sc}}:=\frac1{2\pi}\sqrt{4-x^2}\,\mathbf1_{\{|x|\le2\}}\mathrm dx$, the circular law $\mu_{\mathsf{circ}}:=\mathbf1_{\{|z|\le1\}}\mathrm dz$, and Marčenko−Pastur's law $\mu_{\mathsf{MP}}:=\sqrt\frac{4-s}s\mathbf1_{\{0< s\le 4\}}\mathrm ds$ respectively; namely, $$\{\mu_{\mathsf{sc}}\}=\argmin_{\mu\in\mathcal P(\mathbb R)}\mathcal E^{(2)}_1(\mu),\quad\{\mu_{\mathsf{circ}}\}=\argmin_{\mu\in\mathcal P(\mathbb C)}\mathcal E^{(2)}_{\frac12}(\mu),\quad\text{and}\quad\{\mu_{\mathsf{MP}}\}=\argmin_{\mu\in\mathcal P(\mathbb R_+)}\mathcal E^{(1)}_1(\mu),$$ where $\mathcal P(S)$ denotes the set of probability measures with support in $S$, and $$\mathcal E^{(k)}_b(\mu):=-\Sigma(\mu)+\frac1{kb}\int{|x|}^k\,\mu(\mathrm dx).$$ For a matrix $X_n:=(\xi_{ij})_{1\le i,j\le n}$ of i.i.d. coefficients with mean $0$ and variance $1$, let $$M_n(X):=\frac1{\sqrt n}X_n,\quad W_n(X):=\frac{M_n+M_n^*}{\sqrt2},\quad\text{and}\quad C_n(X):=M_nM_n^*.$$ The aforementioned theorems say that $\mu_{W_n}\Longrightarrow\mu_{\mathsf{sc}}$, that $\mu_{M_n}\Longrightarrow\mu_{\mathsf{circ}}$, and that $\mu_{C_n}\Longrightarrow\mu_{\mathsf{MP}}$ as $n\to\infty$, almost surely (whatever the distribution of the coefficients in $X_n$ may be). The questions we ask, supported by Monte Carlo simulations, are: 1) Does $\mathcal E^{(2)}_1(\mathbb E\mu_{W_n(X)})$ *decrease* to $\mathcal E^{(2)}_1(\mu_{\mathsf{sc}})$? 2) Does $\mathcal E^{(2)}_{\frac12}(\mathbb E\mu_{M_n(X)})$ *decrease* to $\mathcal E^{(2)}_{\frac12}(\mu_{\mathsf{circ}})$? 3) Does $\mathcal E^{(1)}_1(\mathbb E\mu_{C_n(X)})$ *decrease* to $\mathcal E^{(1)}_1(\mu_{\mathsf{MP}})$? In addition to the Monte Carlo simulations, we have positively answered these three questions when $X_n:=(\xi_{ij})_{1\le i,j\le n}$ is i.i.d. with the standard complex Gaussian distribution, in which case $M_n$ is the complex Ginibre ensemble ($\mathsf{Ginibre}$), $W_n$ is the Gaussian Unitary Ensemble ($\mathsf{GUE}$), and $C_n$ is the (square) Laguerre Unitary Ensemble ($\mathsf{LUE}$). In fact, we derived explicit expressions for the logarithmic energy of these three models : $$\begin{aligned}\mathcal E^{(2)}_1(\mathsf{GUE}_n)&=\frac34+\frac12\left(\log n+\gamma+\frac1{2n}-H_n\right),\\[1em]\mathcal E^{(2)}_{\frac12}(\mathsf{Ginibre}_n)&=\mathcal E^{(2)}_1(\mathsf{GUE}_n)+\frac12\sum_{k=n+1}^\infty\frac{4^{-k}\binom{2k}k}{k(k-1)},\\[1em]\mathcal E^{(1)}_1(\mathsf{LUE}_n)&=2\mathcal E^{(2)}_1(\mathsf{GUE}_n).\end{aligned}$$ These three quantities were computed independently, and for the moment we cannot explain the simple relations linking them. Anyway, aside from countless hours spent on computer algebra softwares, this project was a fascinating dive into Euler integrals, orthogonal polynomials, and combinatorial identities involving binomial coefficients and harmonic numbers. [1]: https://doi.org/10.1090/S0894-0347-04-00459-X [2]: https://doi.org/10.1007/s004400050119 [3]: https://doi.org/10.1007/978-3-662-03329-6