This short post is about the singular values of the complex Cauchy or spherical ensemble and its relation to a Jacobi ensemble and to the real Cauchy distribution.
A customary statistical way to define Cauchy distributions is to decide that they are the law of the ratio of two independent random variables following the same centered normal distribution. There is thus a real Cauchy distribution, a complex Cauchy distribution, etc, matrix versions, etc. A more geometric way is to decide that a Cauchy distribution is the image of the uniform distribution on a sphere by a stereographic projection. Both aspects appear naturally in this post.
Complex Cauchy or spherical ensemble. It is the random matrix model \[ AB^{-1} \] where $A$ and $B$ are $n\times n$ independent random matrices with iid entries following the standard complex normal law $\mathcal{N}_{\mathbb{C}}(0,1)=\mathcal{N}_{\mathbb{R}}(0,\frac{1}{2})+\mathrm{i}\mathcal{N}_{\mathbb{R}}(0,\frac{1}{2})$. We could define it using $A^{-1}B$ instead of $AB^{-1}$, but the spectrum is then the same in law. Indeed, $A^{-1}B$ has the same spectrum as $BA^{-1}$, which has the same law as $AB^{-1}$ since $A$ and $B$ are iid.
Since the set of singular $n\times n$ complex matrices is a proper (polynomial) hypersurface of $\mathbb{C}^{n^2}$, and since the law of $A$ (and $B$) is absolutely continuous with respect to the Lebesgue measure on $\mathbb{C}^{n^2}$, it follows that $A$ and $B$ are almost surely invertible.
If $n=1$ then $AB^{-1}$ is a complex random variable following the complex Cauchy law \[ \frac{\mathrm{d}^2z}{\pi(1+|z|^2)^2}=\frac{r\mathrm{d}r\mathrm{d}\theta}{\pi(1+r^2)^2}. \] From this point of view, for $n\geq1$ the law of $AB^{-1}$ can be seen as a matrix version of the Cauchy distribution, hence the name Cauchy ensemble.
The spectrum of $AB^{-1}$ is the determinantal Coulomb gas with density $\propto$ \[ (z_1,\ldots,z_n)\in\mathbb{C}^n \mapsto\prod_{k=1}^n(1+|z_k|^2)^{-(n+1)}\prod_{i < j}|z_i-z_j|^2 =\mathrm{e}^{-(n+1)\sum_{k=1}^nQ(z_k)}\prod_{i< j}|z_i-z_j|^2 \] where $Q=\log(1+\left|\cdot\right|^2)$. Now, the stereographic projection from the unit two-sphere $\mathbb{S}^2=\{x\in\mathbb{R}^3:\|x\|_{\mathbb{R}^3}=1\}$ to $\mathbb{C}\cup\{\infty\}$ associated with the north pole $e_3$ is defined by $T(e_3)=\infty$ and $T(x)=\frac{x_1+\mathrm{i}x_2}{1-x_3}$ if $x\neq e_3$. The image of the gas above by the reverse stereographic projection $T^{-1}$ is the spherical gas on the sphere $\mathbb{S}^2$ with density $\propto$ \[ (s_1,\ldots,s_n)\in(\mathbb{S}^2)^n\mapsto\prod_{i< j}\|s_i-s_j\|^2_{\mathbb{R}^3}. \] Hence the name spherical ensemble. This gas is rotationally invariant. The image by $T$ of the uniform distribution on $\mathbb{S}^2$ is the Cauchy distribution on $\mathbb{C}$. The rotations of $\mathbb{S}^2$ are nothing else but the group $\mathrm{SO}(3)$, and we have \[ \{T\circ R\circ T^{-1}:R\in\mathrm{SO}(3)\} =\mathrm{PSU}(2)\subset\mathrm{PSL}(2,\mathbb{C}). \] This set is formed by Möbius transforms of the form $z\mapsto\frac{az+b}{-\overline{b}z+\overline{a}}$ with $|a|^2+|b|^2=1$. In particular, the spectrum of the Cauchy ensemble is invariant under these transforms.
Squared singular values. The Hermitian positive semidefinite random matrix \[ AB^{-1}(AB^{-1})^*=AB^{-1}B^{-*}A^* \] has the same spectrum as \[ A^*AB^{-1}B^{-*}=(A^*A)(B^*B)^{-1}=VW^{-1}. \] Now $V=A^*A$ and $W=B^*B$ are independent Wishart random matrices, in the Laguerre unitary ensemble. The image of $VW^{-1}$ by the map $X\mapsto (X+I)^{-1}$ is \[ (VW^{-1}+I)^{-1}=W(V+W)^{-1}. \] This random matrix belongs to the Jacobi ensemble. The density of its spectrum is $\propto$ \[ (s_1,\ldots,s_n)\in[0,1]^n\mapsto\prod_{i< j}(s_i-s_j)^2. \] Its equilibrium measure as $n\to\infty$ is the arcsine distribution $\frac{\mathrm{d}s}{\pi\sqrt{s(1-s)}}$ on $[0,1]$.
Using the reverse map $s\mapsto x=s^{-1}-1$ of $x\mapsto s=(x+1)^{-1}$ gives the density $\propto$ \[ (x_1,\ldots,x_n)\in\mathbb{R}_+^n\mapsto\prod_{k=1}^n(1+x_k)^{-2n}\prod_{i< j}(x_i-x_j)^2. \] This heavy-tailed one-sided real Coulomb gas (note that we have $x_k$ and not $x_k^2$) describes the squared singular values of $AB^{-1}$. The equilibrium measure as $n\to\infty$ is the image by the map $s\mapsto s^2$ of the one-sided Cauchy distribution on $\mathbb{R}_+$ \[ \frac{2\mathrm{d}s}{\pi(1+s^2)}. \]
Recall that if $W_1$ and $W_2$ are independent $n\times n$ Wishart random matrices from the $\beta$-Laguerre ensemble with parameters $m_1$ and $m_2$ (sample sizes) then the spectrum of $J:=W_1(W_1+W_2)^{-1}$ is given by the Jacobi ensemble with density $\propto$ \[ (s_1,\ldots,s_n)\in[0,1]^n \mapsto \prod_{k=1}^ns_k^{a_1-p}(1-s_k)^{a_2-p} \prod_{i< j}|s_i-s_j|^\beta \] where $a_i=\frac{\beta}{2}m_i$ and $p=1+\frac{\beta}{2}(n-1)$. In our case, $\beta=2$ and $m_1=m_2=n$, hence $a_1=a_2=p=n$. In multivariate statistics, the ratio $J=W_1(W_1+W_2)^{-1}$ is associated with MANOVA. When $n=m=1$ and $\beta=2$, then $W_1$ and $W_2$ are independent chi-square (special case of Gamma) real random variables, and $J$ is a Beta random variable. The law of $W_2W_1^{-1}=J^{-1}-I$ is the matrix analogue of the Beta-prime or Fisher-Snedecor distribution.
Real Cauchy gases and Cayley transform. The Cayley transform $z\mapsto C(z)=\frac{z-\mathrm{i}}{z+\mathrm{i}}$, which is not the stereographic projection, maps $\mathbb{R}$ to $\mathbb{S}^1$, and more generally the upper half plane to the unit disc (Poincaré to Poincaré). The image by this map of the double-sided real Cauchy gas $\propto$ \[ (x_1,\ldots,x_n)\in\mathbb{R}^n \mapsto\prod_{k=1}^n(1+x_k^2)^{-n}\prod_{i< j}(x_i-x_j)^2 \] is the circular gas on the unit circle $\propto$ \[ (z_1,\ldots,z_n)\in(\mathbb{S}^1)^n \mapsto\prod_{i< j}|z_i-z_j|^2, \] which describes the spectrum of the circular unitary ensemble (Haar unitary matrices). The double-sided real Cauchy gas describes the spectrum of a Hermitian analogue of the Cauchy ensemble, that can be obtained by conjugating with an independent Haar unitary matrix. If $U$ is Haar unitary then $C(U)=(U-\mathrm{i}I)(U+\mathrm{i}I)^{-1}$ is a random Hermitian matrix with spectrum distributed according to the double-sided real Cauchy gas. However that if $A$ and $B$ are independent GUE of size $n\times n$ then $AB^{-1}$ is not Hermitian.
About the Cayley transform, for all $x\in\mathbb{R}$, we have \[ C(x)=\frac{x-\mathrm{i}}{x+\mathrm{i}}=\mathrm{e}^{\mathrm{i}\theta} \] where $\theta\in(-\pi,\pi)$ is such that $x=\tan(\frac{\theta}{2})$, in other words \[ \theta=2\arctan(x). \] The image by $C$ of the uniform distribution on $\mathbb{S}^1$ is the (real) Cauchy distribution on $\mathbb{R}$ \[ \frac{\mathrm{d}x}{\pi(1+x^2)}. \]
Closely related notions. The equilibrium measure of the Jacobi gas, known as the Wachter distribution, has an explicit expression. It contains the arcsine law as a special case, and the semi-circle and Marchenko-Pastur laws as limiting special cases, in relation with the usual Beta/Gamma/Normal and Jacobi/Laguerre/Hermite trilogy.
The Hua-Pickrell gas/ensemble on the unit-circle/unitary-group is obtained by taking the image of the Cauchy gas on the real line by the reverse Cayley transform, at inverse temperature $\beta$.
Further reading.
- On this blog
The spherical ensemble
(2025-10-21) - Peter Forrester
Log-Gases and Random Matrices
Princeton University Press (2010) - Manjunath Krishnapur
Zeros of Random Analytic Functions
PhD manuscript, UC Berkeley (2006) - Alan Edelman and N. Raj Rao
Random matrix theory
Acta Numerica (2005) - Iain Johnstone
High Dimensional Statistical Inference and Random Matrices
Proceedings of the International Congress of Mathematicians (2006) - Paul Bourgade, Ashkan Nikeghbali, and Alain Rouault
Ewens measures on compact groups and hypergeometric kernels
Séminaire de Probabilités XLIII 351-377
Lecture Notes in Mathematics Springer (2006, 2011) - Kartick Adhikari, Nanda K. Reddy, Tulasi R. Reddy, and Koushik Saha
Determinantal point processes in the plane from products of random matrices
Annales de l’Institut Henri Poincaré - Probabilités et Statistiques (2016) - On this blog
Beta/Gamma/Normal and Jacobi/Laguerre/Hermite
(2025-12-21)