CLT. 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?
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 Szegő 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.
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. An important fact to retain is that on the real line, \(-\pi^{-1}\log|\cdot|\) is a fundamental solution of the half-Laplacian, while on the circle, the corresponding chordal logarithmic kernel is a Green kernel with the constant mode removed.
Fourier coefficients of empirical spectral measure. 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 statistic $p_k$, which is a trigonometric polynomial, can then be 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.
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.
Decoupling and normality via Fourier expression of the energy. 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) \[ K(\theta,\phi)=\log\frac{1}{|\mathrm{e}^{\mathrm{i}\theta}-\mathrm{e}^{\mathrm{i}\phi}|}. \] 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 $|z| < 1$, 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.
Haagerup formula. 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-term 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 $L^2(\mu_*)$.
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 $(x,y)\mapsto x$.
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 $[-2,2]$.
The arcsine half-Laplacian. Let $\eta=\mathrm{d}\theta/(2\pi)$ be normalized Haar measure on the circle $\mathbb{S}^1$ and let $L^2_{\mathrm{even}}(\mathbb{S}^1,\eta)$ be the closed subspace of even functions. The map \[ U:L^2([-2,2],\mu_*)\longrightarrow L^2_{\mathrm{even}}(\mathbb{S}^1,\eta), \quad (Uf)(\theta)=f(2\cos\theta), \] 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 $[-2,2]$.
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.
Singular integrals. 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.
Negative Sobolev norm. 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:
- why the arcsine law $\mu_*$ is the equilibrium measure
- why the energy difference is nonnegative
- why the Chebyshev coefficient of degree $n$ is weighted by $1/n$
- why the logarithmic kernel on $[-2,2]$ is naturally associated with a half-Laplacian rather than with a second-order Laplacian.
Variance of linear statistics and Sobolev norm. For a real-valued $f\in L^2(\mathbb{S}^1,\eta)$, set \[ L_n(f)=\sum_{j=1}^nf(\theta_j)-n\widehat f(0) \] where $(\mathrm{e}^{\mathrm{i}\theta_1},\ldots,\mathrm{e}^{\mathrm{i}\theta_n})\sim\mathrm{CUE}_n$ and \[ \widehat f(k)=\int_{\mathbb{S}^1}f(\theta)\mathrm{e}^{-\mathrm{i}k\theta}\mathrm{d}\eta(\theta). \] The random variable $L_n(f)$ is a recentered linear statistic. Its expectation is zero and \[ \operatorname{Var}(L_n(f)) = \sum_{k\ne0}\min(n,|k|)|\widehat f(k)|^2. \] The limit is in $[0,+\infty]$ and is finite iff $f\in H^{1/2}(\mathbb{S}^1)$. In particular, we get \[ \operatorname{Var}(L_n(f)) \xrightarrow[n\to\infty]{} \sum_{k\ne0}|k||\widehat f(k)|^2 ={\|f\|}_{H^{1/2}(\mathbb{S}^1)}^2 \] where \[ H^{1/2}(\mathbb{S}^1)=\bigl\{f\in L^2(\eta):\sum_{k\ne0}|k||\widehat f(k)|^2 < \infty\bigr\}. \] The Sobolev seminorm becomes a norm after quotienting out constants.
Proof. First of all, a direct computation reveals that \[ \mathbb{E}(p_k)=0, \quad \mathbb{E}(p_jp_k)=0, \quad \mathbb{E}(p_j\overline{p_k}) = \mathbf{1}_{j=k}\min(n,k), \quad j,k\geq1. \] These identities are attributed to Persi Diaconis and Steven Evans. Now, if $f$ is a trigonometric polynomial, then $L_n(f)=\sum_{k\ne0}\widehat f(k)p_k$, and the desired variance identity is a finite sum and follows from $\widehat f(-k)=\overline{\widehat f(k)}$. For a general $f\in L^2(\eta)$, it suffices to approximate by the trigonometric polynomial $f_m=\sum_{|k|\le m}\widehat f(k)\mathrm e^{\mathrm{i}k\theta}$. More precisely, at fixed $n$, \[ \mathbb{E}(|L_n(f-f_m)|^2) \leq \mathbb{E}(|\sum_{k=1}^n(f-f_m)(\theta_k)|^2) \leq n^2\|f-f_m\|_{L^2(\eta)}^2\xrightarrow[m\to\infty]{}0. \] Finally, passing to the limit in the polynomial variance formula proves the claim. The series converges, since $\min(n,|k|)\leq n$.
From the CLT for the $p_k$'s to the CLT for linear statistics. For all $m\geq1$, we have \[ (p_1,\ldots,p_m) \xrightarrow[n\to\infty]{\mathrm{d}} (Z_1,\ldots,Z_m) \] where $Z_1,\ldots,Z_m$ are independent with $Z_k\sim\mathcal{N}_{\mathbb{C}}(0,k)$. It follows immediately that for all real-valued $f\in H^{1/2}(\mathbb{S}^1)$ and all $m\geq1$, \[ L_n(f_m)=2\operatorname{Re}\sum_{k=1}^m\widehat f(k)p_k \xrightarrow[n\to\infty]{\mathrm{d}} 2\operatorname{Re}\sum_{k=1}^m\widehat f(k)Z_k. \] Moreover the variance of the limit is $2\sum_{k=1}^mk|\widehat f(k)|^2=\sum_{0 < |k|\leq m}|k||\widehat f(k)|^2$. Furthermore, and as we have already seen \[ \sup_{n\geq1}\mathbb{E}|L_n(f)-L_n(f_m)|^2 \leq \sum_{|k|>m}|k||\widehat f(k)|^2\xrightarrow[m\to\infty]{}0. \] For each $t\in\mathbb{R}$, the difference between the characteristic functions of $L_n(f)$ and $L_n(f_m)$ is bounded by $|t|$ times the square root of this tail. First let $n\to\infty$ with $m$ fixed and then let $m\to\infty$. The resulting characteristic function is $\exp(-\frac{t^2}{2}\|f\|_{H^{1/2}(\mathbb{S}^1)}^2)$, proving the CLT for the linear statistics $L_n(f)$, with a limiting variance given by $\|f\|_{H^{1/2}(\mathbb{S}^1)}^2$.
For finitely many fixed real test functions, the joint version follows by the Cramér-Wold argument, and the covariance is given by \[ \sum_{k\ne0}|k|\widehat f(k)\overline{\widehat g(k)}=\langle f,g\rangle_{H^{1/2}(\mathbb{S}^1)}. \] For real $f,g$ this quantity is real. The proof above concerns fixed test functions, and does not give a result for arbitrary $n$-dependent test functions.
Dual of the energy. Recall that the nonnegative self-adjoint operator $N$ satisfies \[ Ne_k=ke_k, \quad D(N)=\bigl\{f\in L^2(\mu_*):\sum_{k=1}^{\infty}k^2|c_k|^2 < \infty\bigr\},\quad c_k=\langle f,e_k\rangle_{L^2(\mu_*)}. \] Its closed quadratic form is \[ Q(f)=\|N^{1/2}f\|_{L^2(\mu_*)}^2 = \sum_{k=1}^{\infty}k|c_k|^2, \quad D(N^{1/2})=\{f\in L^2(\mu_*):Q(f)<\infty\}. \] The formula $Q(f)=\langle f,Nf\rangle$ is an operator inner product when $f\in D(N)$. For general $f\in D(N^{1/2})$, it is safer to use the quadratic-form notation.
Let $h\in L^2_0(\mu_*)$ be real, with $1+h\ge0$ almost everywhere, and set $\mu=(1+h)\mu_*$. Define \[ \mathcal{E}(\mu)=\iint\log\frac{1}{|x-y|}\mathrm{d}\mu(x)\mathrm{d}\mu(y). \] The integral is absolutely convergent since the logarithmic kernel is in $L^2(\mu_*\otimes\mu_*)$, and the product of the two densities is also in that space. What we did above gives \[ \mathcal{E}(\mu)-\mathcal{E}(\mu_*) =\langle h,N^{-1}h\rangle_{L^2(\mu_*)} =\sum_{k=1}^{\infty}\frac{|h_k|^2}{k}, \qquad h_k=\langle h,e_k\rangle_{L^2(\mu_*)}. \] Actually we have $\mathcal{E}(\mu_*)=0$. Next, for every real $f\in D(N^{1/2})$ and every real $h\in L^2_0(\mu_*)$, whether or not $1+h$ is nonnegative, by Cauchy-Schwarz in the spectral basis, \[ \left|\int fh\mathrm{d}\mu_*\right|^2 \le Q(f)\langle h,N^{-1}h\rangle_{L^2(\mu_*)}. \] The weights $1/k$ in the logarithmic energy and $k$ in the fluctuation variance are dual under the natural pairing $\sum_{k\geq1}(\sqrt{k}c_k)(h_k/\sqrt{k})$, with density perturbations.
Further reading.
- Persi W. Diaconis and Mehrdad M. Shahshahani
On the eigenvalues of random matrices
Journal of Applied Probability (1994) - Persi W. Diaconis and Steven N. Evans
Linear functionals of eigenvalues of random matrices
Transactions of the American Mathematical Society (2001) - Kurt Johansson
On Szegö's asymptotic formula for Toeplitz determinants and generalizations
Bulletin des Sciences Mathématiques (1988) - Kurt Johansson
On random matrices from the compact classical groups
Annals of Mathematics (1997) - Kurt Johansson
On fluctuations of eigenvalues of random Hermitian matrices
Duke Mathematical Journal (1998) - Uffe Haagerup
Seminar notes on free probability
University of Copenhagen, 1998, unpublished - Sylvia Serfaty
Microscopic description of Log and Coulomb gases
Random Matrices, IAS/Park City Math. Series, American Mathematical Society 2019 - Peter J. Forrester
Log-Gases and Random Matrices
Princeton University Press, 2010 - Edward B. Saff and Vilmos Totik
Logarithmic Potentials with External Fields
Springer, 1997. - Michel Ledoux and Ionel Popescu
The one dimensional free Poincaré inequality
Transactions of the American Mathematical Society (2013) - Dan-Virgil Voiculescu
The analogues of entropy and of Fisher's information measure in free probability theory I
Communications in Mathematical Physics (1993)