{"id":23899,"date":"2026-11-14T21:16:16","date_gmt":"2026-11-14T20:16:16","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=23899"},"modified":"2026-11-14T22:24:47","modified_gmt":"2026-11-14T21:24:47","slug":"poincare-inequality-for-ginibre","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2026\/11\/14\/poincare-inequality-for-ginibre\/","title":{"rendered":"Poincar\u00e9 inequality for Ginibre"},"content":{"rendered":"
\"Portrait<\/a>
Henri Poincar\u00e9 (1854 – 1912) in 1887<\/figcaption><\/figure>\n

I gave on Friday November 13 2026 a mini-course on Poincar\u00e9 inequalities for gases, including G$\\beta$E and the Complex Ginibre Ensemble, at the Institut Henri Poincar\u00e9 \ud83d\ude42 for the MEGA seminar<\/a>. An eight-page PDF of lecture notes is available, see [IHP<\/a>]<\/strong> below. This post is devoted to a proof of the optimal Poincar\u00e9 inequality for the complex Ginibre log-gas, after [C<\/a>]<\/strong>, see also [R<\/a>]<\/strong> and [S<\/a>]<\/strong>. Henri Poincar\u00e9 introduced the inequalities bearing his name in [P<\/a>]<\/strong> published in 1890, for his study of partial differential equations.<\/p>\n

The Boltzmann-Gibbs measure.<\/strong> Let $M$ be an $n\\times n$ random matrix with iid complex Gaussian entries of mean $0$ and variance $1\/n$. Its density on $\\mathcal{M}_{n\\times n}(\\mathbb{C})$ is $\\propto\\mathrm{e}^{-n\\mathrm{Tr}(MM^*)}$. Let $\\mu_n$ be the joint distribution of the eigenvalues of $M$. Following Ginibre in [G<\/a>]<\/strong>, \\[ \\mathrm{d}\\mu_n(z) =\\frac{\\mathrm{e}^{-n\\sum_{k=1}^n|z_k|^2}}{c_n}\\prod_{j < k}|z_j-z_k|^2\\mathrm{d}z =\\frac{|V_n(z)|^2}{c_n'}\\mathrm{d}\\gamma_n(z), \\quad V_n(z)=\\prod_{j < k}(z_k-z_j), \\] where $\\gamma_n=\\mathcal{N}_{\\mathbb{C}}(0,\\frac{1}{n}I_n)\\propto\\mathrm{e}^{-n|z|^2}\\mathrm{d}z$ on $\\mathbb{C}^n=\\mathbb{R}^{2n}$ and $c_n'=\\int|V_n|^2\\mathrm{d}\\gamma_n$. This is \\[ \\mu_n\\propto\\mathrm{e}^{-E}, \\quad\\text{with}\\quad E(z)=n|z|^2+\\Phi(z), \\quad\\Phi(z)=2\\sum_{j < k}\\log\\frac{1}{|z_j-z_k|}. \\]<\/p>\n

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  1. For $n\\geq2$, since $\\log\\frac{1}{\\left|\\cdot\\right|}$ on $\\mathbb{R}^2$ has Hessian eigenvalues $\\pm\\left|\\cdot\\right|^{-2}$, $E$ is not convex and the Bakry–\u00c9mery curvature is $-\\infty$. In contrast with the Gaussian Unitary Ensemble, we cannot recover convexity by reordering the coordinates. Moreover $\\log\\frac{1}{\\left|\\cdot\\right|}$ is harmonic on $\\mathbb{C}\\setminus\\{0\\}$ and superharmonic on $\\mathbb{C}$ in the distributional sense since $\\Delta\\log\\left|\\cdot\\right|=2\\pi\\delta_0$, instead of being sub-harmonic, the trace weakening of convexity. The proof below uses holomorphy, which beats convexity in some sense. <\/li>\n
  2. The measure $\\mu_n$ is reversible for a planar Coulomb overdamped Langevin dynamics with generator $L\/n$, where \\begin{align*} L&=\\Delta-\\nabla E\\cdot\\nabla\\\\ &=4\\sum_{i=1}^n\\partial_{z_i}\\partial_{\\bar z_i} -2\\sum_{i=1}^n\\Bigl[ \\Bigl(nz_i-\\sum_{j\\ne i}\\frac{1}{\\bar z_i-\\bar z_j}\\Bigr) \\partial_{\\bar z_i} +\\Bigl(n\\bar z_i-\\sum_{j\\ne i}\\frac{1}{z_i-z_j}\\Bigr) \\partial_{z_i} \\Bigr]. \\end{align*} For $n\\geq2$, this is not the Dyson eigenvalue process of the complex matrix Ornstein-Uhlenbeck diffusion: its eigenvalue quadratic variations involve eigenvector overlaps [BD<\/a>]<\/strong>. This is due to the fact that the matrices from the complex Ginibre ensemble are not normal, in contrast with the Hermitian matrices from the Gaussian Unitary Ensemble. For the same reason, the Poincar\u00e9 inequality for $\\mu_n$ cannot be transferred directly from the Gaussian matrix law by the usual Lipschitz argument: the spectrum is not a Lipschitz function of the entries. <\/li>\n
  3. Since $\\Phi(z+u)=\\Phi(z)$ for all $z$ and all $u\\in F=\\mathbb{C}(1,\\ldots,1)$, the log-gas $\\mu_n$ admits a splitting with a Gaussian factor. Namely it is the image by $(x,y)\\mapsto x(1,\\ldots,1)\/\\sqrt n+y$ of the product measure \\[ \\mathcal{N}_{\\mathbb{C}}\\Bigl(0,\\frac{1}{n}\\Bigr)\\otimes\\nu_n \\] where $\\nu_n$ is the probability measure on $F^\\perp$ with density $\\propto\\mathrm{e}^{-E(y)}$. In other words, the center-of-mass is Gaussian and independent of the recentered gas. For $n\\geq2$, however, the recentered factor $\\nu_n$ is not log-concave, in contrast with the case of the Gaussian Unitary Ensemble. This splitting for the complex Ginibre log-gas was already observed by Robert B. Laughlin in the 1980s in his famous study of the fractional quantum Hall effect, see [L<\/a>]<\/strong>, see also [CL<\/a>]<\/strong>. <\/li>\n<\/ol>\n

    The Poincar\u00e9 inequality.<\/strong> For all $n\\geq1$ and all symmetric and real-valued $f\\in H^1(\\mu_n)$, \\[ \\operatorname{Var}_{\\mu_n}(f)\\leq\\frac{1}{2}\\mathcal{E}_{\\mu_n}(f,f), \\] where $H^1(\\mu_n)$ is the closure of $\\mathcal{C}^\\infty_c(\\mathbb{C}^n)$ with respect to the weighted Sobolev norm, and \\[ \\operatorname{Var}_{\\mu_n}(f) =\\int f^2\\mathrm{d}\\mu_n-\\Bigl(\\int f\\mathrm{d}\\mu_n\\Bigr)^2 \\quad\\text{and}\\quad \\mathcal{E}_{\\mu_n}(f,f) =\\frac{1}{n}\\int_{\\mathbb{C}^n}|\\nabla f|^2\\mathrm{d}\\mu_n. \\] Moreover, equality is achieved if and only if $f$ is a center-of-mass observable: \\[ f=\\Re(aS)+b,\\quad a\\in\\mathbb{C}, b\\in\\mathbb{R}, S=z_1+\\cdots+z_n. \\] In other words the constant $1\/2$ is optimal and the optimal functions are explicit. The symmetry assumption is essential, see the comments below.<\/p>\n

    A proof.<\/strong> We start with the Poincar\u00e9 inequality, and then we consider its optimality. Symmetrization and a smooth symmetric correction of the mean reduce the proof to centered, real, symmetric $f\\in\\mathcal{C}^\\infty_c$. All the $L^2$ spaces that are used are complex. The proof has several steps and involves the following crucial properties:<\/p>\n

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