{"id":22091,"date":"2025-10-21T19:02:57","date_gmt":"2025-10-21T17:02:57","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=22091"},"modified":"2026-05-24T19:12:26","modified_gmt":"2026-05-24T17:12:26","slug":"spherical-ensemble","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2025\/10\/21\/spherical-ensemble\/","title":{"rendered":"Spherical Ensemble"},"content":{"rendered":"

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This post is about the spherical ensemble of random matrices, and some of its properties in potential theory, geometry, and probability. It is at the heart of arXiv:2510.18669<\/a>.<\/p>\n

Coulomb gas.<\/strong> The spherical or Forrester-Krishnapur random matrix ensemble is \\[ M=AB^{-1} \\] where $A$ and $B$ are two independent $n\\times n$ complex Ginibre matrices, namely ${(A_{jk})}_{1\\leq j,k\\leq n}$ and ${(B_{jk})}_{1\\leq j,k\\leq n}$ are iid standard complex normal random variables of law $\\mathcal{N}(0,\\frac{1}{2}\\mathrm{Id}_2)$.<\/p>\n

The law of $M$ is invariant by inversion : $M$ and $M^{-1}$ have the same law. The law of $M$ inherits the biunitary invariance of the complex Ginibre ensemble : invariance by multiplication from the left and from the right by deterministic unitary matrices.<\/p>\n

The random matrix $M$ is a.s. nonnormal and its entries are dependent and heavy-tailed. When $n=1$, then $M$ is a ratio of two independent copies on the standard complex normal distribution, so its law is a complex Cauchy distribution \\[ \\nu=\\kappa(z)\\mathrm{d}z \\quad\\text{with density}\\quad z\\in\\mathbb{C}\\mapsto\\kappa(z)=\\frac{1}{\\pi(1+|z|^2)^2}. \\] The law $\\nu$ is also known as a bivariate Student t distribution in Statistics and as a Barenblatt profile in Analysis of PDEs (fast diffusion equation). It is heavy tailed.<\/p>\n

The spectrum of $M$ is a Coulomb gas on $\\mathbb{C}$ with density $\\varphi_n:\\mathbb{C}^n\\mapsto(0,+\\infty)$ given by \\[ \\varphi_n(z_1,\\ldots,z_n) =c_n\\frac{\\prod_{i < j}|z_i-z_j|^2}{\\prod_{i=1}^n(1+|z_i|^2)^{n+1}} =c_n\\mathrm{e}^{-(n+1)\\sum_{i=1}^nQ(|z_i|)}\\prod_{i < j}|z_i-z_j|^2, \\] where $c_n$ is a normalizing constant, and $Q(|z|)=\\log(1+|z|^2)$.<\/p>\n

By putting this gas in a diagonal matrix and using conjugacy with an independent Haar unitary matrix, this can also be seen as the spectrum of a random normal matrix model, which should not be confused with the random non-normal matrix model $M$.<\/p>\n

Determinantal structure.<\/strong> The planar Coulomb gas above is also a determinantal point process of $n$ particles on $\\mathbb{C}$ endowed with the Lebesgue measure with kernel \\[ (z,w)\\in\\mathbb{C}^2\\mapsto K_n(z,w)=\\sqrt{\\kappa(z)\\kappa(w)} n\\Bigr(\\frac{1+z\\overline{w}}{\\sqrt{(1+|z|^2)(1+|w|^2)}}\\Bigr)^{n-1}. \\] Contrary to certain references, we use here as a background the uniform distribution on the sphere in stereographic coordinates. This means that the joint density writes \\[ \\varphi_n(z_1,\\ldots,z_n)=\\frac{1}{n!}\\det\\bigr[K_n(z_i,z_j)\\bigr]_{1\\leq i,j\\leq n}. \\] The naming comes from the fact that for all $1\\leq k\\leq n$, the marginal density \\[ (z_1,\\ldots,z_k)\\in\\mathbb{C}^k \\mapsto\\varphi_{n,k}(z_1,\\ldots,z_k)=\\int_{\\mathbb{C}^{n-k}}\\varphi_n(z_1,\\ldots,z_n)\\mathrm{d}z_{k+1}\\cdots\\mathrm{d}z_n \\] can be expressed using the kernel as \\[ \\varphi_{n,k}(z_1,\\ldots,z_k) =\\frac{(n-k)!}{n!}\\det\\bigr[K_n(z_i,z_j)\\bigr]_{1\\leq i,j\\leq k}. \\] In particular \\begin{align*} \\varphi_{n,n}&=\\varphi_n\\\\ \\varphi_{n,1}(w)&=\\frac{1}{n}K_n(w,w)=\\kappa(w)\\\\ \\varphi_{n,2}(u,v)&=\\frac{1}{n(n-1)}(K_n(u,u)K_n(v,v)-|K_n(u,v)|^2). \\end{align*} This provides formulas for the mean and variance of linear statistics of the form $L_n(f)=\\sum_{k=1}^nf(\\lambda_k(M))$, namely \\begin{align*} \\mathbb{E}(L_n(f)) &=\\int_{\\mathbb{C}} f(w)K_n(w,w)\\mathrm{d}w=n\\int_{\\mathbb{C}}f(w)\\kappa(w)\\mathrm{d}w\\\\ \\mathrm{Var}(L_n(f)) &=\\int_{\\mathbb{C}}f(w)^2K_n(w,w)\\mathrm{d}w -\\iint_{\\mathbb{C}^2}f(u)f(v)|K_n(u,v)|^2\\mathrm{d}u\\mathrm{d}v\\\\ &=n\\int_{\\mathbb{C}}f(w)^2\\kappa(w)\\mathrm{d}w -\\frac{n^2}{\\pi^2}\\iint_{\\mathbb{C}^2}f(u)f(v)\\frac{|1+u\\overline{v}|^{2(n-1)}}{(1+|u|^2)^{n+1}(1+|v|^2)^{n+1}}\\mathrm{d}u\\mathrm{d}v. \\end{align*}<\/p>\n

Spherical coordinates.<\/strong> Geometrically, the law $\\nu$ is the image of the uniform probability measure on the sphere $\\mathbb{S}^2$ by the (north pole) stereographic projection \\[ T:\\mathbb{S}^2\\subset\\mathbb{R}^3\\to\\mathbb{C}\\cup\\{\\infty\\}. \\] More precisely, for all $x=(x_1,x_2,x_3)\\in\\mathbb{S}^2\\setminus\\{(0,0,1)\\}$ and $z\\in\\mathbb{C}$, \\[ T(x)=\\frac{x_1+\\mathrm{i}x_2}{1-x_3} \\quad\\text{and}\\quad T^{-1}(z) =\\frac{(2 \\Re z, 2 \\Im z,|z|^2-1)}{|z|^2+1}. \\] In other words, this measure is the uniform probability distribution on the sphere written in stereographic coordinates. The image of the spherical ensemble by the inverse stereographic projection $T^{-1}$ is the gas on $\\mathbb {S}^2$ with density with respect to the uniform measure given, up to a multiplicative normalizing constant, by \\[ (x_1,\\ldots,x_n)\\in(\\mathbb{S}^2)^n \\mapsto\\prod_{i < j}\\|x_i-x_j\\|^2_{\\mathbb{R}^3}, \\] hence the name of the spherical ensemble! This can be seen as a perfect two or three dimensional analogue of the circular unitary ensemble (CUE), proportional to \\[ (x_1,\\ldots,x_n)\\in(\\mathbb{S}^1)^n\\subset\\mathbb{C}^n\\mapsto\\prod_{i < j}|x_i-x_j|^2, \\] that describes the spectrum of $n\\times n$ Haar unitary random matrices.<\/p>\n

M\u00f6bius transforms.<\/strong> This gas on $\\mathbb{S}^2$ is invariant by the isometries of $\\mathbb{S}^2$. When $R$ runs over the rotations of $\\mathbb{S}^2$, then $T\\circ R\\circ T^{-1}$ runs over the maps of $\\mathbb{C}\\cup\\{\\infty\\}$ of the form \\[ z\\mapsto\\frac{\\alpha z+\\beta}{-\\overline{\\beta}z+\\overline{\\alpha}}, \\quad (\\alpha,\\beta)\\in\\mathbb{C}^2\\setminus \\{(0,0)\\}. \\] As a consequence, if $Z$ the gas seen on $\\mathbb{C}$, then for all $(\\alpha,\\beta) \\in \\mathbb C^2 \\setminus \\{(0,0)\\}$, \\[ \\frac{\\alpha Z+\\beta}{-\\overline{\\beta}Z+\\overline{\\alpha}} \\overset{\\mathrm{d}}{=} Z. \\] The invariance by scaling suggest to assume that $|\\alpha|^2+|\\beta|^2=1$, and to identify $\\mathbb C \\cup \\{\\infty\\}$ with the projective line $\\mathbb CP^1 = (\\mathbb C^2\\setminus \\{(0,0)\\})\/\\sim$ where $\\sim$ is the complex colinearity equivalence relation. The geometry of this object leads to the Fubini-Study metric on $\\mathbb CP^1$ induced from the Hermitian product of $\\mathbb C^2$.<\/p>\n

Quaternions, SO(3), SU(2), and PSU(2).<\/strong> Algebraically, the group $\\mathrm{SO}(3)$ is isomorphic, by conjugacy by $T$, to the projective subgroup $\\mathrm{PSU(2)}$ of $\\mathrm{SU}(2)$ obtained by taking the quotient by the relation $(\\alpha,\\beta)\\sim(-\\alpha,-\\beta)$, with respect to the parametrization \\[ \\begin{pmatrix}\\alpha & \\beta\\\\ -\\overline{\\beta} &\\overline\\alpha \\end{pmatrix} \\] of $\\mathrm{SU}(2)$. Note that $\\mathrm{SU}(2)$ is isomorphic to the unit sphere of the quaternions $\\mathbb{S}^3\\subset\\mathbb{R}^4$ via $|\\alpha|^2+|\\beta|^2=1$, which leads to another way to link $\\mathrm{SO}(3)$ with $\\mathrm{SU}(2)$. \\[ z\\mapsto\\frac{\\alpha z+\\beta}{-\\overline{\\beta}z+\\overline{\\alpha}}, \\quad \\alpha,\\beta\\in\\mathbb{C},\\quad |\\alpha|^2+|\\beta|^2=1. \\] In particular, for all $z_0\\in\\mathbb{C}$, a M\u00f6bius transform that maps $z_0$ to $0$ is \\[ z\\mapsto M_{z_0}(z)=\\frac{z-z_0}{\\overline{z_0}z+1}. \\] As a consequence, if $Z$ is the gas seen on $\\mathbb{C}$, then for all $z_0\\in\\mathbb{C}$, \\[ M_{z_0}(Z)=\\frac{Z-z_0}{\\overline{z_0}Z+1} \\overset{\\mathrm{d}}{=} Z. \\] Since $M_{z_0}(z_0)=0$ and $M'_{z_0}(z_0)=\\frac{1}{1+|z_0|^2}$, a version of the delta method for random point processes gives that the local behavior of $\\frac{Z-z_0}{1+|z_0|^2}$ near $z_0$ is equal to the one of $Z$ near $0$.<\/p>\n

Kostlan observation and spectral radius.<\/strong> If $Z_n=(Z_{n,1},\\ldots,Z_{n,n})$ is the gas seen as a random vector of $\\mathbb{C}^n$, then the determinantal structure and rotational invariance give \\[ \\{|Z_{n,1}|,\\ldots,|Z_{n,n}|\\}\\overset{\\mathrm{d}}{=}\\{\\xi_{n,1},\\ldots,\\xi_{n,n}\\} \\] where $\\xi_{n,1},\\ldots,\\xi_{n,n}$ are independent and $\\xi_{n,k}$ has density proportional to \\[ x\\geq0\\mapsto x^{2k-1}\\mathrm{e}^{-(n+1)Q(x)}=\\frac{x^{2k-1}}{(1+x^2)^{n+1}}. \\] The random variable $\\xi_{n,k}^2$ has density proportional to \\[ x\\geq0\\mapsto\\frac{x^{k-1}}{(1+x)^{n+1}}. \\] We recognize a Beta prime (or inverse Beta or Beta of the second kind) law of density \\[ x\\geq0\\mapsto\\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)}\\frac{x^{a-1}}{(1+x)^{a+b}}, \\] with $a=k$ and $b=n-k+1$, which is also the law of $B\/(1-B)$ when $B\\sim\\mathrm{Beta}(a,b)$, and also the law of $G_a\/G_b$ were $G_a\\sim\\mathrm{Gamma}(a,\\lambda)$ and $G_b\\sim\\mathrm{Gamma}(b,\\lambda)$ are independent and $\\lambda > 0$ is an arbitrary scale parameter.<\/p>\n

For any fixed $k$, since $\\mathrm{Gamma}(n-k+1,1)=\\mathrm{Exp}(1)^{*(n-k+1)}$, the law of large numbers gives $n\/S_{n,k}\\to1$ a.s. as $n\\to\\infty$, where $S_{n,k}\\sim\\mathrm{Gamma}(n-k+1)$. It follows that \\[ \\frac{n}{\\xi_{n,n-k}^2} \\xrightarrow[n\\to\\infty]{\\mathrm{d}} \\mathrm{Gamma}(k,1), \\] and more generally, the fluctuations of the spectral radius are given by \\[ \\frac{1}{\\sqrt{n}}\\rho(M) =\\frac{1}{\\sqrt{n}}\\max_{1\\leq k\\leq n}|Z_{n,k}| \\overset{\\mathrm{d}}{=} \\frac{1}{\\sqrt{n}}\\max_{1\\leq k\\leq n}\\xi_{n,k} \\xrightarrow[n\\to\\infty]{\\mathrm{d}} \\mathrm{Law}\\Bigr(\\max_{k\\geq1}\\frac{1}{\\sqrt{\\gamma_k}}\\Bigr) \\] where ${(\\gamma_k)}_{k\\geq1}$ are independent with $\\gamma_k\\sim\\mathrm{Gamma}(k,1)$.<\/p>\n

Equilibrium measure.<\/strong> Regarding high dimensional asymptotic analysis, a.s. \\[ \\mu_{M} \\overset{\\mathrm{d}}{=}\\frac{1}{n}\\sum_{k=1}^n\\delta_{Z_{n,k}} \\xrightarrow[n\\to\\infty]{\\text{weak}} \\frac{\\Delta Q(\\left|\\cdot\\right|)}{2\\pi}\\mathrm{d}z =\\frac{\\mathrm{d}z}{\\pi(1+|z|^2)^2} =\\mathrm{\\nu}, \\] The average version is easy to check using logarithmic potential and remains valid non asymptotically. Indeed, for all $z\\in\\mathbb{C}$, $B$ and $A-zB$ are correlated, but by Gaussianity, \\[ (M-z\\mathrm{Id})B =A-zB \\overset{\\mathrm{d}}{=} \\sqrt{1+|z|^2}A. \\] Therefore, for all $z\\in\\mathbb{C}$, in $[-\\infty,+\\infty)$, \\[ \\mathbb{E}\\log|\\det(M-z\\mathrm{Id})|+\\mathbb{E}\\log|\\det(B)| =n\\log\\sqrt{1+|z|^2}+\\mathbb{E}\\log|\\det(A)|. \\] Finally, by applying the operator $\\frac{1}{2\\pi}\\Delta$ in the sense of distributions, we obtain the stunning non-asymptotic formula \\[ \\mathbb{E}\\mu_M=\\frac{\\Delta\\log(1+|z|^2)}{4\\pi}\\mathrm{d}z=\\frac{\\mathrm{d}z}{\\pi(1+|z|^2)^2}=\\mathrm{\\nu}. \\] Alternatively, this formula can be extracted from the determinantal structure, namely \\[ \\mathbb{E}\\mu_M=\\frac{1}{n}K_n(z,z)\\mathrm{d}z=\\kappa(z)\\mathrm{d}z=\\nu. \\] Alternatively, we could simply use the fact that the uniform distribution on $\\mathbb{S}^2$ is the unique distribution on $\\mathbb{S}^2$ invariant by all rotations, together with the fact that its image by the stereographic projection $T$ is precisely $\\nu$ !<\/p>\n

More generally, if $A$ and $B$ are Girko matrices, then $A-zB=\\sqrt{1+|z|^2}C_z$ where $C_z$ is a Girko matrix, but with a law that depends on $z$ in general, except in the Gaussian case. Also the argument above works only in the Ginibre case.<\/p>\n

Coulomb kernel.<\/strong> Let is consider $g:\\mathbb{C}\\times\\mathbb{C}\\mapsto(-\\infty,+\\infty]$ defined for $z,w\\in\\mathbb{C}$ by \\[ g(z,w)=\\log|z-w|-\\tfrac{1}{2}\\log(1+|z|^2)-\\tfrac{1}{2}\\log(1+|w|^2). \\] It is the Coulomb kernel of the two-sphere $\\mathbb{S}^2$ in stereographic coordinates since \\[ \\Delta g(\\cdot,w)\\overset{\\mathcal{D}'}{=}2\\pi(\\delta_w-\\nu). \\] Since $2\\langle z,w\\rangle_{\\mathbb{R}^2}=\\bar{z}w+z\\bar{w}$ for all $z,w\\in\\mathbb{C}^2\\equiv\\mathbb{R}^2$, we get, when $T(x)=z$ and $T(y)=w$, \\begin{align*} 1-\\langle x,y\\rangle_{\\mathbb{R}^3} &=1-\\langle T^{-1}(z),T^{-1}(w)\\rangle_{\\mathbb{R}^3}\\\\ &=1-\\frac{2(\\bar{z}w+z\\bar{w})+(|z|^2-1)(|w|^2-1)}{(1+|z|^2)(1+|w|^2)}\\\\ &=\\frac{2|z-w|^2}{(1+|z|^2)(1+|w|^2)}. \\end{align*} Also, a natural alternative definition of the Coulomb kernel of $\\mathbb{S}^2$ is, for $x,y\\in\\mathbb{S}^2\\subset\\mathbb{R}^3$, \\begin{align*} \\mathfrak{G}(x,y) &= 2\\log\\|x-y\\|_{\\mathbb{R}^3}\\\\ &=\\log\\bigr(1-\\langle x,y\\rangle_{\\mathbb{R}^3}\\bigr)+\\log(2)\\\\ &=2g(z,w)+2\\log(2). \\end{align*} It is symmetric $\\mathfrak{G}(x,y)=\\mathfrak{G}(y,x)$, exhibits logarithmic divergence on the diagonal, and \\[ \\Delta_{\\mathbb{S}^2} \\mathfrak{G}(\\cdot,y) \\overset{\\mathcal{D}'}{=} 4\\pi\\Bigr(\\delta_y -\\frac{\\mathrm{d}x}{4\\pi}\\Bigr) \\] for $y\\in\\mathbb{S}^2$, where $\\Delta_{\\mathbb{S}^2}$ is the Laplace-Beltrami operator on $\\mathbb{S}^2$. Moreover, we have \\[ \\int g(z,w)\\mathrm{d}\\nu(w) =-\\int\\frac{1}{2}\\log(1+|w|^2)\\mathrm{d}\\nu(w) =-\\frac{1}{2}, \\] for $z\\in\\mathbb{C}$ which gives, in particular, for $x\\in\\mathbb{S}^2$, \\[ \\int\\mathfrak{G}(x,y)\\mathrm{d}y =2\\log(2)-1. \\]<\/p>\n

Central Limit Theorem.<\/strong> Since \\[ \\log|\\det(M)|=\\log|\\det(A)|-\\log|\\det(B)| \\] and since $A$ and $B$ are independent and can be Hermitized into complex square Wishart matrices or factorized using the Choleski-Bartlett decomposition, a CLT for $\\log|\\det(M)|$ boils down to a CLT for independent $\\chi^2$ variables. It can also be obtained using the Kostlan observation. More precisely, we get \\[ \\frac{\\log|\\det(M)|}{\\sqrt{\\frac{1}{2}\\log(n)}} \\xrightarrow[n\\to\\infty]{\\mathrm{d}} \\mathcal{N}(0,1). \\] We have $\\mathbb{E}\\log|\\det(M)|=0$ since the law of $M$ is invariant by inversion.<\/p>\n

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