An observation due to Eric Kostlan states that for a large class of radial planar determinantal processes such as the Vandermonde ensembles of random normal-matrix models, the moduli of the particles are independent. The phases remain dependent in general, due to the Vandermonde coupling.
Another observation due to another Eric, Eric Rains, states that 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.
This post is about all that, and the fate of Kostlan and Rains phenomena for general projection determinantal point processes, in relation with Fourier frequencies.
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 rotation-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). This is the model for which the Rains observation was made for the first time by Eric Rains. The Kostlan observation is void here since all the particles have unit modulus.
- 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. This is the model for which the Kostlan observation was made for the first time by Eric Kostlan. Later on, Guillaume Dubach discovered that the Rains phenomenon takes place, and called it "the washing machine effect".
- 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.
Both the circular and spherical ensembles are chordal gases, related to stereographic projection and Cauchy-type ensembles. Moreover, the spherical ensemble and the truncated Haar unitary ensemble are dual of each other in a sense.
Kostlan observation. Since $\mu_n$ is rotation-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 $R_k\propto r^{2(k-1)}\mathrm{d}\nu_n(r)$. Such an equality in distribution of two random sets is in the sense of unordered random vectors, in other words up to an independent uniform permutation of their coordinates. It can also be stated conveniently by using order statistics or point processes as \[ (|Z|_{(1)},\ldots,|Z|_{(n)}) \overset{\mathrm{d}}{=} (R_{(1)},\ldots,R_{(n)}) \quad\text{and}\quad \sum_{k=1}^n\delta_{|Z_k|} \overset{\mathrm{d}}{=} \sum_{k=1}^n\delta_{R_k}. \]
Proof. With $z_k=r_k\mathrm{e}^{\theta_k\mathrm{i}}$, the Vandermonde in the 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 radially symmetric normal-matrix ensembles, 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 \[ N_a= \begin{cases} 1 & \text{if }0\leq a\leq n-1\\ 0 & \text{if }n\leq a\leq m-1 \end{cases}. \] Thus each non-empty block consists of a single point with law proportional to \[ |z|^{\frac{2a}{m}}\,\mathrm{d}\mu_{n,m}(z), \] which is exactly the law of $R_{a+1}^mU_{a+1}$ with the notation used in the Rains observation. At the opposite side, if $m=1$ then $N_0=n$ while $N_a=0$ for all $0 < a \leq m-1$.
Projection DPPs beyond Vandermonde. Vandermonde ensembles describe the joint distribution of the spectrum of random normal-matrix models. They also appear as particular finite-rank projection determinantal point process (DPP). Let us examine the fate of Kostlan and Rains observations in this slightly more general setting. Let \[ \mathrm{d}\mu_n(r\mathrm{e}^{\mathrm{i}\theta}) =\mathrm{d}\nu_n(r)\frac{\mathrm{d}\theta}{2\pi} \] be a rotation-invariant measure on $\mathbb C$. Let $V_n$ be an $n$-dimensional subspace of $L^2(\mu_n)$ which is stable under rotations. The orthogonal projection onto $V_n$ has a kernel \[ K_n(z,w)=\sum_{j=1}^n \phi_j(z)\overline{\phi_j(w)}, \] where $(\phi_1,\ldots,\phi_n)$ is any orthonormal basis of $V_n$. Indeed, for all real valued $f\in L^2(\mu_n)$, \[ K_n(f):=\int_{\mathbb{C}}K_n(\cdot,w)f(w)\mathrm{d}\mu_n(w) =\sum_{j=1}^n\langle\phi_j,f\rangle_{L^2(\mu_n)}\phi_j =\mathrm{proj}_{V_n}(f). \] The associated projection DPP $X_n$ has exactly $n$ points. Their joint (unordered) distribution is \[ \frac{1}{n!}\det{(K_n(z_j,z_k))}_{1\leq j,k\leq n} \prod_{k=1}^n \mathrm{d}\mu_n(z_k) = \frac{1}{n!} \left|\det{(\phi_k(z_j))}_{1\leq j,k\leq n}\right|^2 \prod_{k=1}^n \mathrm{d}\mu_n(z_k). \] The usual Vandermonde case corresponds to $ V_n=\mathrm{span}\{1,z,\ldots,z^{n-1}\}$.
The Andréief determinant identity gives that for all bounded measurable $g:\mathbb{C}\to\mathbb{C}$, \[ \mathbb{E}\Bigl(\prod_{z\in X_n}g(z)\Bigr) =\det\left(\int_{\mathbb{C}}g(z)\phi_j(z)\overline{\phi_k(z)}\mathrm{d}\mu_n(z)\right)_{1\leq j,k\leq n}. \] Let us call it the determinant identity.
The invariance of $V_n$ by rotation means that if $f\in V_n$ then $z\mapsto f(\mathrm{e}^{\mathrm{i}\theta}z)\in V_n$ for all $\theta\in\mathbb{R}$. We can then choose an orthonormal basis of $V_n$ adapted to angular Fourier frequencies: for finitely many integers $\ell$, let $N_\ell\geq0$ be such that $\sum_\ell N_\ell=n$, and write \[ \phi_{\ell,j}(re^{\mathrm{i}\theta}) =u_{\ell,j}(r)e^{\mathrm{i}\ell\theta}, \qquad j\in\{1,\ldots,N_\ell\}. \] For each fixed $\ell$, the radial functions $u_{\ell,1},\ldots,u_{\ell,N_\ell}$ are orthonormal in $L^2(\nu_n)$. In this basis of $V_n$, the kernel $K_n$ admits the decomposition \[ K_n(re^{\mathrm{i}\theta},se^{\mathrm{i}\varphi}) = \sum_\ell e^{\mathrm{i}\ell(\theta-\varphi)} \sum_{p=1}^{N_\ell}u_{\ell,p}(r)\overline{u_{\ell,p}(s)}. \] So the integers $\ell$ are nothing but angular Fourier frequencies. In the Vandermonde case, the active frequencies are $0,1,\ldots,n-1$, each with multiplicity one. The kernel $K_n$ is rotation-invariant in the sense that $K_n(\mathrm{e}^{\mathrm{i}\theta}z,\mathrm{e}^{\mathrm{i}\theta}w)=K_n(z,w)$ for all $z,w\in\mathbb{C}$, $\theta\in\mathbb{R}$.
Kostlan for projection DPPs. Taking $g(z)=h(|z|)$ gives \[ \int_{\mathbb C}h(|z|) \phi_{\ell,p}(z)\overline{\phi_{\ell',p'}(z)}\mathrm{d}\mu_n(z) =0 \quad\text{if}\quad \ell\neq \ell'. \] Indeed, the angular integral contains $\int_0^{2\pi}e^{\mathrm{i}(\ell-\ell')\theta}\frac{\mathrm{d}\theta}{2\pi}$. Therefore, the matrix in the determinant identity is block diagonal with one block for each angular frequency $\ell$: \[ \mathbb{E}\Bigl(\prod_{z\in X_n}h(|z|)\Bigr) = \prod_\ell \det\left( \int_0^\infty h(r)u_{\ell,p}(r)\overline{u_{\ell,q}(r)}\mathrm{d}\nu_n(r) \right)_{1\leq p,q\leq N_\ell}. \] For each $\ell$, let $Y_\ell$ be the projection DPP on $\mathbb R_+$ with reference measure $\nu_n$ and kernel \[ L_\ell(r,s) = \sum_{p=1}^{N_\ell}u_{\ell,p}(r)\overline{u_{\ell,p}(s)}. \] Then the last identity says precisely that \[ \{|z|:z\in X_n\} \overset{\mathrm d}{=} \bigcup_\ell Y_\ell, \] where the processes $Y_\ell$ are independent. This is a sort of weakened Kostlan observation in projection-DPP form: the moduli split according to angular frequency. The independence of all the radii à la Kostlan occurs when every active angular frequency has multiplicity one, namely $N_\ell\leq1$ for every $\ell$. In that case $Y_\ell$ consists of a single point with density $|u_{\ell,1}(r)|^2\mathrm{d}\nu_n(r)$. For instance, in the Vandermonde case, $u_{j,1}(r)=r^j/h_j$, $h_j^2=\int_0^\infty r^{2j}\mathrm{d}\nu_n(r)$, and we recover the Kostlan observation $R_j\propto r^{2(j-1)}\mathrm{d}\nu_n(r)$.
On the contrary, if $N_\ell > 1$ for some $\ell$, then the corresponding radii form a radial projection DPP and are not independent in general.
Rains for projection DPPs. Fix $m\geq1$, and take now $g(z)=h(z^m)$. In the determinant identity, with the Fourier basis, the matrix entry between $\phi_{\ell,p}$ and $\phi_{\ell',p'}$ is \[ \int_0^\infty\int_0^{2\pi} h(r^m e^{\mathrm{i}m\theta}) u_{\ell,p}(r)\overline{u_{\ell',p'}(r)} e^{\mathrm{i}(\ell-\ell')\theta} \frac{\mathrm{d}\theta}{2\pi}\mathrm{d}\nu_n(r). \] The function $\theta\mapsto h(r^m e^{\mathrm{i}m\theta})$ is invariant under $\theta\mapsto\theta+2\pi/m$. Hence the angular integral vanishes unless $ \ell\equiv \ell'\pmod m$. Thus the determinant is block diagonal, not by the exact angular frequency $\ell$, but by its residue class modulo $m$.
For $a\in\{0,\ldots,m-1\}$, let $X_{n,a}^{(m)}$ be independent projection DPPs associated with the subspaces $V_{n,a}^{(m)}=\mathrm{span}\{\phi_{\ell,p}:\ell\equiv a\pmod m,\ 1\leq p\leq N_\ell\}$. Their kernels are \[ K_{n,a}^{(m)}(z,w) = \sum_{\ell\equiv a\pmod m} \sum_{j=1}^{N_\ell} \phi_{\ell,j}(z)\overline{\phi_{\ell,j}(w)}. \] The block factorization of the finite determinant gives \[ \{z^m:z\in X_n\} \overset{\mathrm d}{=} \bigcup_{a=0}^{m-1} \{z^m:z\in X_{n,a}^{(m)}\}, \] where the processes on the right-hand side are independent. This is the Rains superposition for projection-DPP: the map $z\mapsto z^m$ separates angular frequencies modulo $m$.
Full independence of the powered points occurs if every residue block has dimension at most one: $\dim V_{n,a}^{(m)}\leq1$ for all $a\in\{0,\ldots,m-1\}$. This condition is automatic in the Vandermonde case when $m\geq n$ since the frequencies $0,1,\ldots,n-1$ have distinct residues modulo $m$, leading to the Rains phenomenon \[ \{Z_1^m,\ldots,Z_n^m\}\overset{\mathrm d}{=}\{R_1^mU_1,\ldots,R_n^mU_n\}, \] with independent $R_j$ as in the Kostlan observation, and independent uniform phases $U_j$.
If $m < n$ in the Vandermonde case, the residue blocks need not be one dimensional. Writing as before $N_a=\#\{0\leq j\leq n-1:j\equiv a\pmod m\}$, the $a$-th block is generated by the monomials $z^a,z^{a+m},z^{a+2m},\ldots$ which still lie between degrees $0$ and $n-1$. If $\mu_{n,m}$ is the image of $\mu_n$ by $z\mapsto z^m$, then the image of this block is the Vandermonde-type projection DPP of size $N_a$ with reference measure \[ |z|^{\frac{2a}{m}}\mathrm{d}\mu_{n,m}(z). \] Equivalently its joint distribution is proportional to \[ \prod_{j < k}|z_j-z_k|^2\prod_{k=1}^{N_a}|z_k|^{\frac{2a}{m}}\mathrm{d}\mu_{n,m}(z_k). \] This recovers the Rains observation. The Vandermonde case is special : each angular frequency has multiplicity one. A general rotation-invariant projection DPP has the same modular block structure, but its blocks may have dimension larger than one.
About Eric (James) Kostlan. The University of California, Berkeley, Department of Mathematics lists him as a 1985 Ph.D. recipient, with the dissertation Statistical Complexity of Numerical Linear Algebra and Stephen Smale as his advisor. He was publishing mathematical work in the early 1990s, including papers in random matrix theory and random polynomial theory. One 1992 paper lists him as affiliated with the Department of Mathematics at the University of Hawaii, and the well-known Edelman–Kostlan paper, titled How Many Zeros of a Random Polynomial Are Real?, appeared in 1995. It seems that he left academic mathematics around 2000 to work in the network and security engineering industry. He spent six years at Symantec/VERITAS and more than 18 years at Cisco Systems. His LinkedIn profile lists him as a Technical Marketing Engineer at Cisco Systems. His name remains very much alive in random polynomial theory and random algebraic geometry.
About Eric M. Rains. He is an Emeritus Professor of Mathematics at the California Institute of Technology. A highly precocious student, he entered Case Western Reserve University at the age of 14 and graduated at 17 with a bachelor degrees in computer science and physics and a master degree in mathematics. He then obtained a Certificate of Advanced Study in Mathematics from the University of Cambridge and a Ph.D. in Mathematics from Harvard University in 1995, with the dissertation Topics in Probability on Compact Lie Groups, under the supervision of Persi Diaconis. After positions at the Center for Communications Research in Princeton and AT&T Labs, he was a professor at the University of California, Davis, and then at Caltech, where he became professor emeritus in 2023. His work spans quantum information theory and coding theory, random matrix theory, special functions, noncommutative geometry, and number theory.
Personal. My own interest in these topics came when I have realized that the Kostlan observation leads to a probabilistic proof of the Gumbel fluctuation at the edge of the complex Ginibre ensemble. This led to a joint work with Sandrine Péché, more than ten years ago.
Further reading
- 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) - 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 - Djalil Chafaï and Sandrine Péché
A note on the second order universality at the edge of Coulomb gases on the plane
Journal of Statistical Physics (2014) - Guillaume Dubach
Powers of Ginibre eigenvalues
Electronic Journal of Probability (2018)
