This short post is about a remarkable non-asymptotic probabilistic phenomenon: when folding a radial determinantal process with a power map, the dependence disappears as soon as the number of foldings is at least the number of particles. It is an exact identity in distribution which simply boils down to the modular identity \[ \sum_{r=0}^{m-1} \mathrm{e}^{2\pi\frac{rq}{m}\mathrm{i}} = m\mathbf{1}_{q\equiv 0\pmod m}. \]
Circular and spherical ensembles as low dimensional chordal gases. Let us denote by $\mathbb{S}^{d-1}=\{x\in\mathbb{R}^d:|x|_{\mathbb{R}^d}^2=x_1^2+\cdots+x_d^2=1\}$ the unit sphere of $\mathbb{R}^d$, $d\geq2$. The chordal gas at inverse temperature $\beta > 0$ is the probability measure on $(\mathbb{S}^{d-1})^n$, $n\geq1$, with density with respect to the product surface measure proportional to \[ (v_1,\ldots,v_n)\in(\mathbb{S}^{d-1})^n \mapsto \prod_{j < k}|v_j-v_k|^\beta_{\mathbb{R}^d}. \] Its image by the north pole stereographic projection \[ T:v\in\mathbb{S}^{d-1}\to T(v)=\frac{(v_1,\ldots,v_{d-1})}{1-v_d}\in\mathbb{R}^{d-1}\cup\{\infty\}, \] is the Cauchy ensemble on $(\mathbb{R}^{d-1})^n$ with density proportional to \[ (w_1,\ldots,w_n)\in(\mathbb{R}^{d-1})^n \mapsto\frac{\prod_{j < k}|w_j-w_k|_{\mathbb{R}^{d-1}}^\beta}{\prod_{k=1}^n(1+|w_k|_{\mathbb{R}^{d-1}}^2)^{d-1+\frac{\beta}{2}(n-1)}}. \] The key formulas for reverse stereographic projection and half chordal distance are \[ T^{-1}(w)=\Bigl(\frac{2w}{1+|w|^2},\frac{|w|^2-1}{1+|w|^2}\Bigr) \quad\text{and}\quad \frac{|T^{-1}(w_1)-T^{-1}(w_2)|}{2} =\frac{|w_1-w_2|}{\sqrt{1+|w_1|^2}\sqrt{1+|w_2|^2}}. \]
When $d=2$ and $\beta=2$, this is the circular unitary ensemble. This matches the joint distribution of the eigenvalues of a random uniform $n\times n$ unitary matrix following the uniform probability measure on the unitary group (normalized Haar measure). Its image by the stereographic projection is the Cauchy ensemble on $\mathbb{R}^n$ with density proportional to \[ (x_1,\ldots,x_n)\in\mathbb{R}^n \mapsto\frac{\prod_{j < k}(x_j-x_k)^2}{\prod_{k=1}^n(1+x_k^2)^n}. \]
When $d=3$ and $\beta=2$, this is the spherical ensemble. Its image by the stereographic projection is the Cauchy ensemble on $\mathbb{C}^n$ with density proportional to \[ (z_1,\ldots,z_n)\in\mathbb{C}^n \mapsto\frac{\prod_{j < k}|z_j-z_k|^2}{\prod_{k=1}^n(1+|z_k|^2)^{n+1}}. \] This matches the joint distribution of the eigenvalues of random $n\times n$ matrix $AB^{-1}$ where $A$ and $B$ are independent $n\times n$ random matrices from the complex Ginibre ensemble (iid standard complex Gaussian entries). This time the random matrix model is on the Euclidean side rather than on the spherical side. In both cases, the random matrix model is on the side of complex numbers : circle for circular, plane for spherical.
Rains theorem. For all $m\geq n$, the image measure of the circular ensemble by the map \[ \Phi_m:(z_1,\ldots,z_n)\mapsto (z_1^m,\ldots,z_n^m) \] is the product measure $\mu^{\otimes n}$ where $\mu$ is the uniform probability measure on the unit circle $\mathbb{S}^1$. In terms of random variables, if $U$ is an $n\times n$ Haar unitary random matrix then for all integer $m\ge n$, \[ \mathrm{spec}(U^m) \overset{\mathrm{d}}=\{\xi_1,\ldots,\xi_n\}, \] where $\xi_1,\ldots,\xi_n$ are independent uniform random variables on $\mathbb{S}^1$. This follows immediately from the chordal formula, by computing the Jacobian of the change of variable.
More generally, for all integer $m\ge1$, \[ \mathrm{spec}(U^m)\overset{\mathrm d}= \bigcup_{r=0}^{m-1}\mathrm{spec}(U_r) \] where $U_0,\ldots,U_{m-1}$ are independent Haar unitary random matrices, with $U_r$ of dimension $N_r\times N_r$ with $N_r=\#\{\ell\in\{0,\ldots,n-1\}:\ell\equiv r\pmod m\}$. When $m\ge n$, then $N_r$ is equal to $0$ or $1$, and the equality in distribution statement reduces to the first one.
Dubach theorem. For all $m\geq n$, if $A$ is an $n\times n$ complex Ginibre matrix and if $B$ is an independent copy, then \[ \mathrm{spec}((AB^{-1})^m) \overset{\mathrm d}= \{\xi_1,\ldots,\xi_n\} \] where this time $\xi_1,\ldots,\xi_n$ are independent random variables on $\mathbb{C}$, with uniform phases, and \[ |\xi_k|^{2/m}\sim\mathrm{BetaPrime}(k,n+1-k). \] Here again a proof consists in computing the Jacobian of the change of variable from the chordal gas. Contrary to the circular ensemble case, here the radii may not be i.i.d.
As observed by Dubach, the decoupling under $\Phi_m$ remains available beyond the spherical ensemble, for general two-dimensional determinantal processes with radially symmetric potential. Moreover, for all $m\ge1$, a Rains-type superposition theorem is available, just like for the circular ensemble.
The result includes the Kostlan observation on the independence of the radii when $m=1$.
Further reading
- Guillaume Dubach
Powers of Ginibre eigenvalues
Electronic Journal of Probability (2018) - J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and Bálint Virág
Determinantal processes and independence
Probability Surveys (2006) - J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and Bálint Virág
Zeros of Gaussian Analytic Functions and Determinantal Point Processes
University Lecture Series, American Mathematical Society 2009 - Eric Kostlan
On the spectra of Gaussian matrices
Linear Algebra and its Applications (1992) - Manjunath Krishnapur
From random matrices to random analytic functions
The Annals of Probability (2009) - Eric M. Rains
High powers of random elements of compact Lie groups
Probability Theory and Related Fields (1997) - Eric M. Rains
Images of eigenvalue distributions under power maps
Probability Theory and Related Fields (2003)