An observation due to Eric Kostlan (Ph.D. Berkeley 1985, with Stephen Smale) states that for a radial planar determinantal process, the moduli of the particles are independent. The phases remain dependent in general, due to the Vandermonde coupling.
This post is about another observation due to another Eric, Eric Rains (Ph.D. Harvard 1995, with Persi Diaconis): both phases and radii become independent after taking the image of the gas by a power larger than or equal to the number of particles. In other words, the Vandermonde coupling disappears by folding. This decoupling phenomenon boils down to an elementary modular identity, which is already beautiful.
The model. In the sequel we consider the probability measure $P_n$ on $\mathbb{C}^n$ of the form \[ \frac{1}{Z_n}\prod_{j < k}|z_j-z_k|^2\prod_{k=1}^n\mathrm{d}\mu_n(z_k) \] where $Z_n$ is the normalizing constant and where $\mu_n$ is a rotationally invariant measure on $\mathbb{C}$ with finite moments up to order $2(n-1)$, namely \[ \int_{\mathbb{C}}|z|^{2k}\mathrm{d}\mu_n(z) < \infty,\quad 0\leq k\leq n-1. \] The gas $P_n$ describes the joint distribution of the complex spectrum of several random matrix ensembles. Here are some well known examples.
- Circular unitary ensemble : $\mu_n$ is the normalized arc-length measure on the unit circle $\mathbb{S}^1=\{z\in\mathbb{C}:|z|=1\}$. Then $P_n$ matches the joint distribution of the eigenvalues of a random uniform $n\times n$ unitary matrix $U$ following the uniform probability measure on the unitary group (normalized Haar measure).
- Complex Ginibre ensemble : $\mu_n$ is $\mathcal{N}_{\mathbb{C}}(0,1/n)$ with density proportional to \[ z\mapsto\mathrm{e}^{-n|z|^2}. \] Then $P_n$ matches the joint distribution of the eigenvalues of a complex Ginibre matrix : $n\times n$ with iid $\mathcal{N}_{\mathbb{C}}(0,1/n)$ entries.
- Spherical ensemble : $\mu_n$ has density proportional to \[ z\mapsto(1+|z|^2)^{-(n+1)}. \] Then $P_n$ matches the joint distribution of the eigenvalues of the random $n\times n$ matrix $AB^{-1}$ where $A$ and $B$ are independent $n\times n$ complex Ginibre matrices. It is also the image of a chordal two-sphere gas by the stereographic projection, hence the name.
- Truncated unitary ensemble : $\mu_n$ has density proportional to \[ z\mapsto(1-|z|^2)^{N-n-1}\mathbf{1}_{|z|\leq1},\quad N > n. \] Then $P_n$ matches the joint distribution of the eigenvalues of the main $n\times n$ block of an $N\times N$ Haar unitary matrix. This can be seen as a sort of complex Jacobi gas.
- Product of complex Ginibre ensembles : $\mu_n$ has density proportional to \[ z\mapsto w_N(z) = G_{0,N}^{N,0} \!\left( \left. \begin{matrix} -\\ 0,\ldots,0 \end{matrix} \right||z|^2 \right) \] the Meijer $G$ function. Then $P_n$ matches the joint distribution of the eigenvalues of the product of $N\geq1$ iid $n\times n$ complex Ginibre random matrices.
Kostlan observation. Since $\mu_n$ is rotationally invariant, it is proportional to $\mathrm{d}\nu_n(r)\frac{1}{2\pi}\mathrm{d}\theta$, equivalently, if $X\propto\mu_n$ then its modulus and phase are independent and distributed according to $\nu_n$ and the uniform law on $[0,2\pi]$. Now, if $(Z_1,\ldots,Z_n)\sim P_n$, then \[ \{|Z_1|,\ldots,|Z_n|\}\overset{\mathrm{d}}{=}\{R_1,\ldots,R_n\} \] where $R_1,\ldots,R_n$ are independent random variables with $R_k\propto r^{2(k-1)}\mathrm{d}\nu_n(r)$. This equality in distribution of two random sets is in the sense of order statistics or point processes respectively, namely up to an independent random permutation: \[ (|Z|_{(1)},\ldots,|Z|_{(n)}) \overset{\mathrm{d}}{=} (R_{(1)},\ldots,R_{(n)}) \quad\text{and}\quad \sum_{k=1}^n\delta_{|Z_k|} \overset{\mathrm{d}}{=} \sum_{k=1}^n\delta_{R_k}. \]
Proof. With $z_k=r_k\mathrm{e}^{\theta_k\mathrm{i}}$, the Vandermonde in density of $P_n$ writes \begin{align*} \prod_{j < k}|z_j-z_k|^2 &=\prod_{j < k}(z_j-z_k)\overline{\prod_{j < k}(z_j-z_k)}\\ &=\det({(z_j^{k-1})}_{1\leq j,k\leq n})\det({(\overline{z_j}^{k-1})}_{1\leq j,k\leq n})\\ &=\Bigl(\sum_{\sigma\in\Sigma_n}(-1)^{\varepsilon(\sigma)}\prod_{j=1}^nz_j^{\sigma(j)-1}\Bigr) \Bigl(\sum_{\sigma'\in\Sigma_n}(-1)^{\varepsilon(\sigma')}\prod_{j=1}^n\overline{z_j}^{\sigma'(j)-1}\Bigr)\\ &=\Bigl(\sum_{\sigma,\sigma'\in\Sigma_n}(-1)^{\varepsilon(\sigma)+\varepsilon(\sigma')} \prod_{j=1}^nz_j^{\sigma(j)-1}\overline{z_j}^{\sigma'(j)-1}\Bigr)\\ &=\Bigl(\sum_{\sigma,\sigma'\in\Sigma_n}(-1)^{\varepsilon(\sigma)+\varepsilon(\sigma')} \prod_{j=1}^nr_j^{\sigma(j)-1+\sigma'(j)-1}\mathrm{e}^{(\sigma(j)-\sigma'(j))\theta_j\mathrm{i}}\Bigr). \end{align*} Integrating the density of $P_n$ with respect to the phases $\theta_1,\ldots,\theta_n$ forces $\sigma=\sigma'$ and gives \[ \sum_{\sigma\in\Sigma_n}\prod_{j=1}^nr_j^{2(\sigma(j)-1)}\prod_{k=1}^n\mathrm{d}\nu_n(r_k) =\mathrm{per}({(r_j^{2(k-1)})}_{1\leq j,k\leq n})\prod_{k=1}^n\mathrm{d}\nu_n(r_k). \] This is the law of $(R_{\sigma(1)},\ldots,R_{\sigma(n)})$ for a random uniform permutation $\sigma$ independent of $(R_1,\ldots,R_n)$ with $R_1,\ldots,R_n$ independent with $R_k\propto r^{2(k-1)}\mathrm{d}\nu_n(r)$, in other words the density of the order statistics $(R_{(1)},\ldots,R_{(n)})$ up to a multiplicative factor $1/n!$.
Rains observation. For all $m\geq n$, if $(Z_1,\ldots,Z_n)\sim P_n$, then \[ \{Z_1^m,\ldots,Z_n^m\} \overset{\mathrm{d}}{=} \{R_1^mU_1,\ldots,R_n^mU_n\} \] where $R_1,\ldots,R_n,U_1,\ldots,U_n$ are independent with $U_1,\ldots,U_n$ uniformly distributed on the unit circle $\mathbb{S}^1=\{z\in\mathbb{C}:|z|=1\}$, and $R_k\propto r^{2(k-1)}\mathrm{d}\nu_n(r)$ as in the Kostlan observation.
Proof. We would like to parametrize $z_k=r_k\mathrm{e}^{\theta_k\mathrm{i}}$ with $z_k^m=\rho_k\mathrm{e}^{\varphi_k\mathrm{i}}$. This gives $r_k=\rho_k^{1/m}$ and $\theta_k=\varphi_k/m+(2\pi)a_k/m$ with $a_k\in\{0,\ldots,m-1\}$, in other words \[ z_k=r_kx_k\eta^{a_k} \quad\text{where}\quad x_k=\mathrm{e}^{\frac{\varphi_k}{m}\mathrm{i}} \quad\text{and}\quad \eta=\mathrm{e}^{\frac{2\pi}{m}\mathrm{i}}. \] Thus we have to compute the following sum over the extra parameters $a_k$: \[ \sum_{a_1,\ldots,a_n=0}^{m-1}\prod_{j < k}|r_jx_j\eta^{a_j}-r_kx_k\eta^{a_k}|^2. \] Now \begin{align*} \prod_{j < k}|r_jx_j\eta^{a_j}-r_kx_k\eta^{a_k}|^2 &=\prod_{j < k}(r_jx_j\eta^{a_j}-r_kx_k\eta^{a_k}) \prod_{j < k}\overline{(r_jx_j\eta^{a_j}-r_kx_k\eta^{a_k})}\\ &=\sum_{\sigma,\sigma'\in\Sigma_n}(-1)^{\varepsilon(\sigma)+\varepsilon(\sigma')} \prod_{j=1}^nr_j^{\sigma(j)-1+\sigma'(j)-1}x_j^{\sigma(j)-\sigma'(j)}\eta^{a_j(\sigma(j)-\sigma'(j))}. \end{align*} A summation over $a_1,\ldots,a_n$ leads to the evaluation of \[ \sum_{a_1,\ldots,a_n=0}^{m-1}\prod_{j=1}^n\eta^{a_j(\sigma(j)-\sigma'(j))} =\prod_{j=1}^n\sum_{a_j=0}^{m-1}\eta^{a_j(\sigma(j)-\sigma'(j))}. \] Now a standard modular identity (see below) gives that \[ \sum_{a_j=0}^{m-1}\eta^{a_j(\sigma(j)-\sigma'(j))} =m\mathbb{1}_{\sigma(j)-\sigma'(j)\equiv0\pmod m}. \] But $|\sigma(j)-\sigma'(j)|\leq n-1 < m$, hence $\sigma(j)-\sigma'(j)\equiv0\pmod m$ is equivalent to $\sigma(j)=\sigma'(j)$. Therefore the product over $1\leq j\leq n$ vanishes unless $\sigma=\sigma'$. Therefore the summation over $a_1,\ldots,a_n$ gives finally \[ \sum_{a_1,\ldots,a_n=0}^{m-1} \left|\Delta(r_1x_1\eta^{a_1},\ldots,r_nx_n\eta^{a_n})\right|^2 =m^n\sum_{\sigma\in\Sigma_n}\prod_{j=1}^nr_j^{2(\sigma(j)-1)} =m^n\mathrm{per}({(r_j^{2(k-1)})}_{1\leq j,k\leq n}). \] Finally, the Jacobian of $(\theta_1,\ldots,\theta_n)\mapsto(\varphi_1, \ldots,\varphi_n)$ contributes $m^{-n}$. We get then the symmetrized law of the independent variables $R_1^mU_1,\ldots,R_n^mU_n$.
Modular identity. The Rains observation boils down to the modular identity \[ S_{m,q}:=\sum_{a=0}^{m-1} \mathrm{e}^{\frac{2\pi q}{m}a \mathrm{i}} = m\mathbf{1}_{q\equiv 0\pmod m},\quad m\in\mathbb{N}, q\in\mathbb{Z}. \] Its proof is as follows, after introducing the $m$-th root of unity $\omega=\mathrm{e}^{\frac{2\pi}{m}\mathrm{i}}$:
- If $q\equiv0\pmod m$, then $q=km$ for some $k\in\mathbb{Z}$, so $\omega^q=1$, thus $S_{m,q}=m$.
- If $q\not\equiv0\pmod m$, then $\omega^q\neq1$ and then \[ S_{m,q}=1+\omega^q+\cdots+\omega^{(m-1)q}=\frac{1-(\omega^q)^m}{1-\omega^q}. \] But $(\omega^q)^m=\omega^{qm}=(\omega^m)^q=1$, thus $S_{m,q}=0$.
This is also the standard orthogonality identity of the characters of the cyclic group $\mathbb{Z}/m\mathbb{Z}$. More precisely, we have indeed $S_{m,q}=\langle\chi_0,\chi_q\rangle_{\ell^2_{\mathbb{C}}(\mathbb{Z}/m\mathbb{Z}\to\mathbb{C})}$ where $\chi_q(a):=\mathrm{e}^{\frac{2\pi}{m}qa\mathrm{i}}$. Behind Fourier or harmonic analysis, it is just basic trigonometric identities: \[ \sum_{a=0}^{m-1} \cos\Bigl(\frac{2\pi q}{m}a\Bigr) = m\mathbf{1}_{q\equiv 0\pmod m} \quad\text{and}\quad \sum_{a=0}^{m-1} \sin\Bigl(\frac{2\pi q}{m}a\Bigr) = 0. \]
Rains superposition. Actually Eric Rains has also elucidated the case $m < n$ for the circular unitary ensemble, and his observation was generalized later on, to arbitrary radial planar determinantal processes, by Guillaume Dubach. More precisely, for all $m\geq1$, if $(Z_1,\ldots,Z_n)\sim P_n$, then the following superposition in distribution holds: \[ \{Z_1^m,\ldots,Z_n^m\} \overset{\mathrm{d}}{=} \bigcup_{a=0}^{m-1}S_a \] where $S_0,\ldots,S_{m-1}$ are the sets formed by the coordinates of independent random vectors $X^{(0)},\ldots,X^{(m-1)}$ of $\mathbb{C}^{N_0},\ldots,\mathbb{C}^{N_{m-1}}$, with \[ N_a=\#\{\ell\in\{0,\ldots,n-1\}:\ell\equiv a\pmod m\}, \] the distribution of $X^{(a)}$ being proportional to \[ \prod_{j < k}|z_j-z_k|^2\prod_{k=1}^{N_a}|z_k|^{\frac{2a}{m}}\mathrm{d}\mu_{n,m}(z_k) \] where $\mu_{n,m}$ is the image measure of $\mu_n$ by the power map $z\mapsto z^m$.
Note that when $m\geq n$, then \[ N_a= \begin{cases} 1, & 0\leq a\leq n-1,\\ 0, & n\leq a\leq m-1. \end{cases} \] Thus each non-empty block consists of a single point with law proportional to \[ |z|^{2a/m}\,\mathrm{d}\mu_{n,m}(z), \] which is exactly the law of $R_{a+1}^mU_{a+1}$ with the notation used in the Rains observation.
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) - 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)