{"id":23357,"date":"2026-09-25T19:00:18","date_gmt":"2026-09-25T17:00:18","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=23357"},"modified":"2026-09-25T19:20:47","modified_gmt":"2026-09-25T17:20:47","slug":"aspects-of-cue","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2026\/09\/25\/aspects-of-cue\/","title":{"rendered":"Aspects of CUE"},"content":{"rendered":"
\"Photo<\/a>
Mehrdad Mirshams Shahshahani Ph.D. Berkeley 1970, with Ichiro Satake. An early explorer of CLTs for CUE. He should not be confused with his brother Siavash Mirshams Shahshahani, Ph.D. Berkeley 1969 with Stephen Smale.<\/figcaption><\/figure>\n

CLT.<\/strong> This post is about a famous high-dimensional phenomenon for the unitary group $\\mathbb{U}(n)$. More precisely, let $U_n$ be a random $n\\times n$ unitary matrix following the uniform distribution, or normalized Haar measure. For any integer $k\\in\\mathbb{Z}$, let us define \\[ p_k=\\mathrm{Tr}(U_n^k)=\\overline{p_{-k}}. \\] Then, for all $m\\geq1$, \\[ (p_1,\\ldots,p_m) \\xrightarrow[n\\to\\infty]{\\mathrm{d}} \\mathcal{N}_{\\mathbb{C}}(0,1)\\otimes\\cdots\\otimes\\mathcal{N}_{\\mathbb{C}}(0,m) \\] where $\\mathcal{N}_{\\mathbb{C}}(0,k)=\\mathcal{N}_{\\mathbb{R}}(0,\\frac{1}{2}k)+\\mathrm{i}\\mathcal{N}_{\\mathbb{R}}(0,\\frac{1}{2}k)$ has density $z\\in\\mathbb{C}\\mapsto\\frac{1}{\\pi k}\\mathrm{e}^{-\\frac{|z|^2}{k}}$. This is a joint central limit theorem (CLT) for the linear spectral statistics $p_1,\\ldots,p_m$. It expresses asymptotic normality and independence. How to guess it, how to understand it?<\/p>\n

There are plenty of ways to prove this result. An early proof is the one given by Persi Diaconis and Mehrdad Shahshahani, using representation theory. Kurt Johansson provided an alternative proof based on Fourier analysis and the strong Szeg\u0151 theorem on the determinant of Toeplitz matrices. He also gave yet another proof based on a loop equation and integration by parts for the log-gas of the joint law of the eigenvalues.<\/p>\n

But giving a proof is not always the best way to understand intuitively the phenomenon. The aim of this post is to explain how Fourier analysis reveals a hidden normal product structure behind the Boltzmann--Gibbs measure of the joint law of the eigenvalues. In a second step, we explain how this is related to a Haagerup formula, the Chebyshev polynomials of the first kind, the half-Laplacian, and singular integrals.<\/p>\n

Fourier coefficients of empirical spectral measure.<\/strong> The eigenvalues of $U_n$ belong to the unit circle $\\mathbb{S}^1=\\{z\\in\\mathbb{C}:|z|=1\\}$, and the law of $U_n$ is known as the Circular Unitary Ensemble (CUE). The joint distribution of the eigenvalues of $U_n$ is \\[ (z_1,\\ldots,z_n)\\in(\\mathbb{S}^1)^n \\mapsto\\frac{1}{Z_n}\\prod_{1\\leq j < k\\leq n}|z_j-z_k|^2\\mathrm{d}z_1\\cdots\\mathrm{d}z_n \\] where $Z_n$ is the normalizing constant and where $\\mathrm{d}z_1\\cdots\\mathrm{d}z_n$ stands for the uniform probability measure on $(\\mathbb{S}^1)^n$. In terms of phases, we get, with $z_j=\\mathrm{e}^{\\mathrm{i}\\theta_j}$, \\[ (\\theta_1,\\ldots,\\theta_n)\\in[0,2\\pi]^n \\mapsto\\frac{1}{Z_n'}\\prod_{1\\leq j < k\\leq n}|\\mathrm{e}^{\\mathrm{i}\\theta_j}-\\mathrm{e}^{\\mathrm{i}\\theta_k}|^2\\mathrm{d}\\theta_1\\cdots\\mathrm{d}\\theta_n \\] where $\\mathrm{d}\\theta_1\\cdots\\mathrm{d}\\theta_n$ stands for the uniform probability measure on $[0,2\\pi]^n$. The linear spectral statistics $p_k$, which is a trigonometric polynomial, can be then rewritten as \\[ p_k = \\sum_{j=1}^n\\mathrm{e}^{\\mathrm{i}k\\theta_j} = \\int\\mathrm{e}^{\\mathrm{i}k\\theta}\\mathrm{d}\\nu_n(\\theta) = \\widehat{\\nu}_n(k) \\quad\\text{where}\\quad \\nu_n = \\sum_{j=1}^n\\delta_{\\theta_j} \\] is the unnormalized empirical distribution of the phases of the eigenvalues.<\/p>\n

Also the CLT of interest expresses the following Fourier phenomenon: for all $m\\geq1$, \\[ (\\widehat{\\nu}_n(1),\\ldots,\\widehat{\\nu}_n(m)) \\xrightarrow[n\\to\\infty]{\\mathrm{d}} \\mathcal{N}_{\\mathbb{C}}(0,1)\\otimes\\cdots\\otimes\\mathcal{N}_{\\mathbb{C}}(0,m). \\] In other words, in the high-dimensional limit, the Fourier coefficients or Fourier modes of $\\nu_n$ decouple and become normal, with variance equal to their number or frequency.<\/p>\n

Decoupling and normality via Fourier expression of the energy.<\/strong> Let us see the joint spectral distribution of $U_n$ as a Boltzmann--Gibbs measure, namely \\[ \\prod_{j < k}|\\mathrm{e}^{\\mathrm{i}\\theta_j}-\\mathrm{e}^{\\mathrm{i}\\theta_k}|^2 =\\mathrm{e}^{-E_n(\\theta_1,\\ldots,\\theta_n)} \\] where, for pairwise distinct eigenvalues, the energy $E_n$ is given by \\[ E_n(\\theta_1,\\ldots,\\theta_n) = -\\sum_{j\\neq k}\\log|\\mathrm{e}^{\\mathrm{i}\\theta_j}-\\mathrm{e}^{\\mathrm{i}\\theta_k}| = \\iint_{\\neq} K(\\theta,\\phi)\\mathrm{d}\\nu_n(\\theta)\\mathrm{d}\\nu_n(\\phi) \\] which involves the kernel (logarithmic or 2D Coulomb, restricted on the circle) \\[ K(\\theta,\\phi)=-\\log|\\mathrm{e}^{\\mathrm{i}\\theta}-\\mathrm{e}^{\\mathrm{i}\\phi}|. \\] Now, recall that for all $t\\notin 2\\pi\\mathbb{Z}$, \\[ \\log|1-\\mathrm{e}^{\\mathrm{i}t}| = -\\sum_{k=1}^{\\infty}\\frac{\\cos(kt)}{k} \\] The series converges, but in general it does not converge absolutely. This comes from $\\log(1-z)=-\\sum_{n=1}^{\\infty}\\frac{z^n}{n}$ for $\\lvert z\\rvert < 1$, with $z=r\\mathrm{e}^{\\mathrm{i}t}$ and $r\\nearrow1$. It follows that \\[ -\\log |\\mathrm{e}^{\\mathrm{i}\\theta}-\\mathrm{e}^{\\mathrm{i}\\phi}| = \\sum_{k=1}^\\infty\\frac{\\cos(k(\\theta-\\phi))}{k}. \\] As a consequence, this expansion of the kernel away from the diagonal gives \\begin{align*} E_n(\\theta_1,\\ldots,\\theta_n) &=\\sum_{k\\geq 1}\\iint_{\\neq}\\frac{\\mathrm{e}^{\\mathrm{i}k\\theta}\\mathrm{e}^{-\\mathrm{i}k\\phi}+\\mathrm{e}^{-\\mathrm{i}k\\theta}\\mathrm{e}^{\\mathrm{i}k\\phi}}{2k}\\mathrm{d}\\nu_n(\\theta)\\mathrm{d}\\nu_n(\\phi)\\\\ &=\\sum_{k\\geq 1}\\frac{|\\widehat{\\nu}_n(k)|^2-n}{k}\\\\ &=\\sum_{k\\geq 1}\\frac{|p_k|^2-n}{k}. \\end{align*} The sums are limits of partial sums in increasing $k$; the subtraction of $n$ removes the diagonal contribution and is configuration-independent at each cutoff. This remarkable quadratic expression of the energy suggests that the Fourier coefficients of $\\nu_n$, which are the $p_k$'s, become asymptotically independent and normal, with $p_k$ of variance $k$. Of course, this is only a heuristic, since we are implicitly using a non-linear change of variable to pass from the eigenvalues to the empirical measure of their phases. The Fourier coefficients are constrained, and the image of the reference measure is not a product Lebesgue measure.<\/p>\n

Haagerup formula.<\/strong> We start from the following trigonometric observation: \\begin{align*} |1-\\mathrm{e}^{\\mathrm{i}t}| &=\\sqrt{(1-\\cos(t))^2+\\sin(t)^2}\\\\ &=\\sqrt{2-2\\cos(t)}\\\\ &=\\sqrt{2-2(2\\cos(\\tfrac{t}{2})^2-1)}\\\\ &=2|\\sin(\\tfrac{t}{2})|. \\end{align*} Now using $\\log|1-\\mathrm{e}^{\\mathrm{i}t}|=-\\sum_{k\\geq1}\\frac{\\cos(kt)}{k}$ together with \\[ \\cos(a+b)-\\cos(a-b)=-2\\sin(a)\\sin(b) \\] and \\[ \\cos(k(a-b))+\\cos(k(a+b))=2\\cos(ka)\\cos(kb), \\] we get \\begin{align*} \\log|2\\cos(\\theta)-2\\cos(\\phi)| &=\\log\\Bigl(2\\Bigl|\\sin\\Bigl(\\frac{\\theta-\\phi}{2}\\Bigr)\\Bigr|\\Bigr) +\\log\\Bigl(2\\Bigl|\\sin\\Bigl(\\frac{\\theta+\\phi}{2}\\Bigr)\\Bigr|\\Bigr)\\\\ &=-2\\sum_{k=1}^\\infty\\frac{\\cos(k\\theta)\\cos(k\\phi)}{k}. \\end{align*} Writing $x=2\\cos(\\theta)$, $y=2\\cos(\\phi)$, $T_n(\\cos(\\theta))=\\cos(n\\theta)$, gives, for all $x\\neq y\\in[-2,2]$, \\[ \\log|x-y| = -2\\sum_{n=1}^{\\infty} \\frac{1}{n}T_n\\Bigl(\\frac{x}{2}\\Bigr) T_n\\Bigl(\\frac{y}{2}\\Bigr). \\] The series is convergent and is generally only conditionally convergent. This is a famous formula, attributed to Uffe Haagerup, which was also known in classical potential theory. It is well known that ${(T_n)}_{n\\geq0}$ are the Chebyshev polynomials of the first kind which satisfy the three terms recurrence relation $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ with initial conditions $T_0\\equiv1$ and $T_1(x)=x$. We have \\[ \\int T_m\\Bigl(\\frac{x}{2}\\Bigr)T_n\\Bigl(\\frac{x}{2}\\Bigr)\\mathrm{d}\\mu_*(x) =\\begin{cases} 1 & \\text{if $m=n=0$}\\\\ \\frac{1}{2} & \\text{if $m=n\\geq1$}\\\\ 0 & \\text{if $m\\neq n$} \\end{cases} \\quad\\text{where}\\quad \\mu_*=\\frac{\\mathbf{1}_{x\\in [-2,2]}}{\\pi\\sqrt{4-x^2}}\\mathrm{d}x \\] is the arcsine law on $[-2,2]$. In particular, the sequence of polynomials ${(e_n)}_{n\\geq0}$ defined by $e_0(x)\\equiv1$ and $e_n(x)=\\sqrt{2}T_n(x\/2)$ for $n\\geq 1$ is an orthonormal basis of $L^2(\\mu_*)$.<\/p>\n

The arcsine distribution $\\mu_*$ is the image measure of the uniform distribution on $[0,\\pi]$ by the map $\\theta\\mapsto2\\cos(\\theta)$. It is also the image measure of the uniform distribution on the centered circle of radius $2$ by the coordinate projection map $(x,y)\\mapsto x$.<\/p>\n

The arcsine distribution $\\mu_*$ is the equilibrium measure on $[-2,2]$, in other words the minimizer of the logarithmic energy \\[ \\mathcal{E}(\\mu)=\\iint\\log\\frac{1}{|x-y|}\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y) \\] over the set of probability measures $\\mu$ supported in $[-2,2]$.<\/p>\n

The arcsine half-Laplacian.<\/strong> Let $m=\\mathrm{d}\\theta\/(2\\pi)$ be normalized Haar measure on the circle $\\mathbb{T}$ and let $L^2_{\\mathrm{even}}(\\mathbb T,m)$ be the closed subspace of even functions. The map \\[ U:L^2([-2,2],\\mu_*)\\longrightarrow L^2_{\\mathrm{even}}(\\mathbb T,m), \\quad (Uf)(\\theta)=f(2\\cos\\theta), \\] is unitary, in particular it is a linear isometry. The half-Laplacian \\[ \\sqrt{-\\Delta}=\\sqrt{-\\partial_\\theta^2} \\] on the circle is the Fourier multiplier operator defined by \\[ \\sqrt{-\\Delta}\\mathrm{e}^{\\mathrm{i}n\\theta} = |n|\\mathrm{e}^{\\mathrm{i}n\\theta}. \\] Conjugating with respect to $U$, we get positive self-adjoint operators on $L^2([-2,2],\\mu_*)$: \\[ J = U^{-1}(-\\Delta)U \\quad\\text{and}\\quad N = U^{-1}\\sqrt{-\\Delta}U = \\sqrt{J}. \\] On polynomials, the first operator is actually the Chebyshev or Jacobi operator \\[ Jf(x) = -(4-x^2)f''(x)+xf'(x) \\] The action on Chebyshev polynomials of the first kind is given by \\[ JT_n\\bigl(\\frac{x}{2}\\bigr) = n^2T_n\\bigl(\\frac{x}{2}\\bigr) \\quad\\text{and}\\quad NT_n\\bigl(\\frac{x}{2}\\bigr) = nT_n\\bigl(\\frac{x}{2}\\bigr). \\] We call $N$ the Chebyshev counting operator. It is the half-Laplacian obtained from the Neumann Laplacian on $[0,\\pi]$ under $x=2\\cos(\\theta)$. It should not be confused with the restriction of the Euclidean fractional Laplacian on $[-2,2]$.<\/p>\n

The spectral kernel of $N^{-1}$ on the orthogonal complement $L^2_0([-2,2],\\mu_*)$ of the constants in $L^2([-2,2],\\mu_*)$ is given by \\[ \\sum_{n=1}^{\\infty}\\frac{e_n(x)e_n(y)}{n} = 2\\sum_{n=1}^{\\infty}\\frac{T_n\\bigl(\\frac{x}{2}\\bigr)T_n\\bigl(\\frac{y}{2}\\bigr)}{n} \\] which is identified by the Haagerup formula with $-\\log|x-y|$ off the diagonal. Moreover, since $\\sum_{n=1}^{\\infty}\\frac{1}{n^2} < \\infty$, the spectral series defines a Hilbert--Schmidt kernel and converges in $L^2([-2,2]^2,\\mu_*\\otimes\\mu_*)$. This gives that for all $f\\in L^2_0([-2,2],\\mu_*)$, \\[ (N^{-1}f)(x) = \\int\\log\\frac{1}{|x-y|}f(y)\\mathrm{d}\\mu_*(y) =\\Bigl(\\log\\frac{1}{\\left|\\cdot\\right|}*(f\\mu_*)\\Bigr)(x). \\] The equality holds in $L^2$ and pointwise whenever the integral and the corresponding spectral series admit pointwise interpretations. This can be rewritten in terms of Schwartz distributions as follows: \\[ N_x\\bigl(-\\log\\lvert x-y\\rvert\\bigr) = \\delta_y-1, \\] where the identity is understood relative to $\\mu_*$: when paired with a test function $f$, the right-hand side gives $f(y)-\\int f(x)\\mathrm{d}\\mu_*(x)$. The subtraction of $1$ removes the component in the kernel of $N$. This is the arcsine$[-2,2]$ analogue of the fact that $-\\frac{1}{\\pi}\\log\\left|\\cdot\\right|$ is the fundamental solution of the half-Laplacian $\\sqrt{-\\Delta}$ on $\\mathbb{R}$, which is a fractional Laplacian.<\/p>\n

Singular-integrals.<\/strong> For a smooth function $g$ on the circle, we have \\[ \\sqrt{-\\Delta}g(\\theta) = \\frac{1}{4\\pi}\\operatorname{p.v.} \\int_0^{2\\pi}\\frac{g(\\theta)-g(\\varphi)}{\\sin^2((\\theta-\\varphi)\/2)}\\mathrm{d}\\varphi. \\] Pairing the points $\\varphi$ and $-\\varphi$, and then using $x=2\\cos(\\theta)$, $y=2\\cos(\\varphi)$, gives, for sufficiently regular $f$ and $x\\in(-2,2)$, \\[ Nf(x) = \\operatorname{p.v.}\\int\\frac{(f(x)-f(y))(4-xy)}{(x-y)^2}\\mathrm{d}\\mu_*(y). \\] Indeed, $\\frac{1}{\\sin^2((\\theta-\\varphi)\/2)} + \\frac{1}{\\sin^2((\\theta+\\varphi)\/2)} = \\frac{4(4-xy)}{(x-y)^2}$ and by symmetrization, \\[ \\langle f,Nf\\rangle_{L^2(\\mu_*)} = \\frac12\\iint\\frac{|f(x)-f(y)|^2(4-xy)}{(x-y)^2}\\mathrm{d}\\mu_*(x)\\mathrm{d}\\mu_*(y) \\geq0. \\] This formula is the arcsine$[-2,2]$ analogue of the Riesz kernel formula of the half-Laplacian $\\sqrt{-\\Delta}$ on $\\mathbb{R}$ seen as a fractional Laplacian.<\/p>\n

Negative Sobolev norm.<\/strong> Suppose first that $\\mu=f\\mu_*$ is a probability measure with $f\\in L^2(\\mu_*)$. Since the logarithmic potential of $\\mu_*$ vanishes on $[-2,2]$, we get \\begin{align*} \\mathcal{E}(\\mu)-\\mathcal{E}(\\mu_*) &=\\iint\\log\\frac{1}{|x-y|}(f(x)-1)(f(y)-1)\\mathrm{d}\\mu_*(x)\\mathrm{d}\\mu_*(y)\\\\ &=\\left\\langle f-1,N^{-1}(f-1)\\right\\rangle_{L^2(\\mu_*)}\\\\ &=\\left\\lVert\\sqrt{N^{-1}}(f-1)\\right\\rVert_{L^2(\\mu_*)}^2. \\end{align*} By approximation, the corresponding identity extends to finite-energy probability measures on $[-2,2]$, with the right-hand side interpreted as the squared negative Sobolev norm of the signed measure $\\mu-\\mu_*$. The formula above explains simultaneously:<\/p>\n

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