{"id":19014,"date":"2023-12-12T12:12:30","date_gmt":"2023-12-12T11:12:30","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=19014"},"modified":"2026-04-20T19:01:19","modified_gmt":"2026-04-20T17:01:19","slug":"mccarthy-multimatrices-and-log-gases","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2023\/12\/12\/mccarthy-multimatrices-and-log-gases\/","title":{"rendered":"McCarthy multimatrices and log-gases"},"content":{"rendered":"
\"Photo<\/a>
Freeman Dyson (1923 - 1920) - Great explorer of links between random matrix ensembles and log-gases<\/figcaption><\/figure>\n

The McCarthy multimatrix ensemble of random matrices.<\/strong> For all integers $n\\geq1$ and $d\\geq1$, let $\\mathbb{M}_{n,d}$ be the set of $d$-tuples $(M_1,\\ldots,M_d)$ of $n\\times n$ Hermitian matrices such that
\n\\[
\nM_pM_q=M_qM_p\\quad\\text{for all $1\\leq p,q\\leq d$.}
\n\\] We equip this hypersurface with the trace of the Gaussian distribution, namely
\n\\[
\n(M_1,\\ldots,M_d)
\n\\mapsto\\frac{1}{Z_{n,d}}
\n\\mathrm{e}^{-\\sum_{k=1}^d\\mathrm{Tr}(M_k^2)}\\mathrm{d}M
\n\\] where $\\mathrm{d}M$ is the trace on $\\mathbb{M}_{n,d}$ of the (product) Lebesgue measure on $d$-uples of $n\\times n$ Hermitian matrices, and $Z_{n,d}$ is the normalizing constant. Since the Hermitian matrices $M_1,\\ldots,M_d$ commute, they are diagonalizable in the same orthonormal basis, namely there exists a single $n\\times n$ unitary matrix $U$ and real $n\\times n$ diagonal matrices $D_1,\\ldots,D_d$ carrying the eigenvalues of $M_1,\\ldots,M_d$ respectively such that
\n\\[
\nU(M_1,\\ldots,M_d)U^*
\n=(D_1,\\ldots,D_d).
\n\\] The eigenvectors couple the eigenvalues : $\\lambda_i\\in\\mathbb{R}^d$ is an eigenvalue of $(M_1,\\ldots,M_d)$ when there exists $u\\in\\mathbb{R}^n$, $u\\neq0$, such that $M_pu=\\lambda_{i,p}u$ for all $1\\leq p\\leq n$. The computation of the Jacobbian of the spectral change of variable provides the following remarkable fact: the joint law of the eigenvalues $\\lambda_1,\\ldots,\\lambda_n$ of $(M_1,\\ldots,M_d)$ has probability density function
\n\\[
\n(\\lambda_1,\\ldots,\\lambda_n)\\in(\\mathbb{R}^d)^n
\n\\mapsto\\frac{1}{Z_{n,d}}
\n\\mathrm{e}^{-\\sum_{i=1}^n\\|\\lambda_i\\|^2}
\n\\prod_{1\\leq i< j\\leq n}\\|\\lambda_i-\\lambda_j\\|^2
\n\\mathrm{d}\\lambda
\n\\] where $\\mathrm{d}\\lambda$ is the Lebesgue measure on $(\\mathbb{R}^d)^n$, $\\|x\\|^2:=x_1^2+\\cdots+x_d^2$ is the squared Euclidean norm of $\\mathbb{R}^d$, and $Z_{n,d}$ is the normalizing constant. When $d=1$, we recover the usual formula for the Gaussian Unitary Ensemble (GUE), studied notably by Freeman Dyson. Thus the model generalizes the GUE into a ``MacCarthy Unitary Ensemble'' (MUE). When $d=2$, we obtain the formula for the spectrum of the complex Ginibre ensemble, providing a novel interpretation of this formula in terms of spectrum of two commuting Hermitian matrices!<\/p>\n

The log-gas picture and the analogue of the Wigner theorem.<\/strong> The law above is a log-gas of $n$ particles on $\\mathbb{R}^d$ with quadratic external field, namely
\n\\[
\n\\mathrm{e}^{-\\sum_{i=1}^n\\|\\lambda_i\\|^2}
\n\\prod_{1\\leq i< j\\leq n}\\|\\lambda_i-\\lambda_j\\|^2
\n=\\exp\\Bigr(n\\int V\\mathrm{d}\\mu_n+\\iint_{\\neq}W\\mathrm{d}\\mu_n^{\\otimes 2}\\Bigr)
\n\\] where
\n\\[
\nV(x):=\\|x\\|^2,\\quad
\nW(x,y):=\\log\\frac{1}{\\|x-y\\|},\\quad
\n\\mu_n:=\\frac{1}{n}\\sum_{i=1}^n\\delta_{\\lambda_i}.
\n\\] What is the behavior of the empirical measure $\\mu_n$ under this law? By computing the correlation functions, or by using the Laplace method, we get that after scaling by $n^{-1\/2}$, the empirical measure in high dimension $n$ tends to the equilibrium measure
\n\\[
\n\\mu_{\\mathrm{eq}}
\n=\\arg\\min_\\mu\\Bigr(\\int V\\mathrm{d}\\mu+\\iint W\\mathrm{d}\\mu^{\\otimes 2}\\Bigr)
\n\\] where the minimum runs over the set of probability measures on $\\mathbb{R}^d$. It turns out that these equibrium measures were already computed, quite recently for $d\\geq3$, namely
\n\\[
\n\\mu_{\\mathrm{eq}}
\n=
\n\\begin{cases}
\n\\frac{1}{\\pi} \\sqrt{2-x^2} \\mathbf{1}_{|x|\\leq\\sqrt{2}}
\n& \\text{if $d=1$}\\\\
\n\\frac{1}{\\pi} \\mathbf{1}_{\\|x\\|\\leq1}
\n& \\text{if $d=2$}\\\\
\n\\frac{3}{2\\pi^2}\\frac{1}{\\sqrt{\\frac{2}{3}-\\|x\\|^2}}\\mathbf{1}_{\\|x\\|\\leq\\sqrt{\\frac{2}{3}}}
\n&\\text{if $d=3$}\\\\
\n\\sigma_{S^{d-1}(\\frac{1}{\\sqrt{2}})}
\n& \\text{if $d\\geq 4$}
\n\\end{cases}
\n\\] where $\\sigma_{S^{d-1}(R)}$ is the uniform distribution on the sphere $S^{d-1}:=\\{x\\in\\mathbb{R}^d:\\|x\\|=R\\}$ of radius $R$. Also $\\mu_{\\mathrm{eq}}$ is radially symmetric and its one-dimensional projections are semi-circle distributions. We have an analogue or generalization of the Wigner theorem for the McCarthy multimatrix Ensemble. The Wigner theorem for GUE corresponds to $d=1$.<\/p>\n

More generally, if we incorporate a parameter $\\beta>0$, then the unique minimizer of the functional
\n\\[
\n \\mu\\mapsto
\n \\mathcal{E}_\\beta(\\mu)=\\int\\|x\\|^2\\mathrm{d}\\mu+\\frac{\\beta}{2}\\iint\\log\\frac{1}{\\|x-y\\|}\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)
\n\\] is
\n\\[
\n \\mu_{\\mathrm{eq}}^\\beta =\\arg\\min\\mathcal{E}_\\beta
\n =\\begin{cases} \\displaystyle
\n \\frac{d}{\\beta\\pi^{\\frac{d}{2}}\\Gamma(2-\\frac{d}{2})}\\Bigl(\\frac{\\beta}{d}-\\|x\\|^2\\Bigr)_+^{\\frac{2-d}{2}}\\mathrm{d}x
\n & \\text{if $d\\in\\{1,2,3\\}$}\\\\
\n \\displaystyle \\sigma_{S^{d-1}(\\frac{\\sqrt{\\beta}}{2})}
\n & \\text{if $d\\geq4$}
\n \\end{cases}
\n\\] It is the semicircle distribution $\\frac{2}{\\beta\\pi}\\sqrt{(\\beta-\\|x\\|^2)_+}\\mathrm{d}x$ for $d=1$, the uniform distribution on a disc $\\frac{2}{\\beta\\pi}\\mathbf{1}_{\\|x\\|^2\\leq\\frac{\\beta}{2}}\\mathrm{d}x$ for $d=2$, and a radial arcsine or Riesz-Barenblatt distribution for $d=3$. The cases $d\\in\\{1,2\\}$ are well known, while the cases $d\\geq3$ and the condensation phenomenon for $d\\geq4$ is a recent discovery. The distribution $\\mu_{\\mathrm{eq}}^\\beta$ becomes more and more singular when $d$ increases, and ceases to have a density for $d=4$. The logarithmic repulsion is more and more weak, making the cost of being at the surface more desirable than a spread over the ball.<\/p>\n

Note.<\/strong> Lydia Giacomin is studying these questions as a side project during her PhD.<\/p>\n

Further reading.<\/strong><\/p>\n