# Month: September 2017

Mes collègues me demandent souvent en quoi consiste ma mission de vice-président chargé du numérique, et à quel point je parviens à mener en parallèle une activité de recherche. À Paris-Dauphine et comme dans un certain nombre d’autres universités françaises, les vice-présidents bénéficient d’une décharge totale d’enseignement. Cela dit, j’ai choisi de continuer à enseigner un peu, par goût pour l’enseignement. Ma mission de vice-président consiste à élaborer et à mener la politique de l’établissement en matière de systèmes d’information et de numérique, sur tous les plans et pour tous les acteurs de l’université : étudiants, enseignants, chercheurs, administratifs. En quelques chiffres, le numérique à Paris-Dauphine, c’est un budget d’environ 4 M€ par an, 45 personnes à temps plein à la direction des systèmes d’information, et plus de 140 systèmes informatiques faisant appel à plus de 20 technologies différentes. Le caractère transversal de la mission fait que je dois participer à beaucoup de réunions et intervenir dans beaucoup de projets collectifs. À titre d’exemple, voici l’emploi du temps de mes deux premiers jours de rentrée :

Lundi :
09h00-12h00 Réunion hebdomadaire du comité exécutif de Paris-Dauphine (politique de l’établissement)
12h00-13h30 Déjeuner de travail avec le directeur des systèmes d’information et la directrice de la communication (ENT, communication interne)
13h30-17h30 Réunion avec le directeur des systèmes d’information et le directeur de la bibliothèque (schéma directeur numérique)
17h30-19h30 Debriefing avec le directeur des systèmes d’information

Mardi :
09h30-11h45 Réunion avec le vice-président chargé du numérique et le DSI de PSL (masters, candidatures, diplomations, …)
12h00-13h00 Réunion de présentation du M1 de maths (dont je suis responsable) aux ~140 étudiants inscrits
13h30-15h30 Réunion du conseil du labo CEREMADE (dont je fais partie)
15h30-17h30 Réunion au Centre d’Innovation Pédagogique avec la société Blackboard (plateforme d’apprentissage en ligne).

Les jours suivants ont été un peu moins chargés, mais le morcellement ne m’a pas permis d’avoir une activité de recherche. Fort heureusement, je parviens à ménager au fil des semaines quelques jours successifs dédiés à la recherche. Les voyages à l’étranger peuvent aussi constituer une manière de sauvegarder du temps, mais les dossiers urgents ne peuvent pas attendre et sont alors traités via courriel, téléphone, et Skype. La mission de vice-président chargé du numérique est passionnante, même si elle peut s’avérer difficile. C’est l’occasion d’apprendre, comprendre, façonner, et de mettre à profit le goût des mathématiciens pour l’efficacité, la rigueur, et la résolution de problèmes.

This post is about a joint work arXiv:1610.00980 in collaboration with Adrien Hardy and Mylène Maïda. We study the non-asymptotic behavior of Coulomb gases in dimension ${d\geq2}$. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our work is inspired by the one of Mylène Maïda and Édouard Maurel-Segala (2014) for one-dimensional log-gases, itself inspired by large deviations techniques. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. It allows to recover, extend, and simplify previous results by Nicolas Rougerie and Sylvia Serfaty (2016).

Motivation. Our main motivation is the remakable concentration of measure phenomenon which holds for the complex Ginibre ensemble: if you sample a square random matrix in high dimension with independent and identically distributed complex Gaussian entries of mean zero and variance one then the empirical measure of the eigenvalues of this random matrix will be very close to the uniform distribution on a centered disc with radius equal to the square root of the dimension. It turns out that the law of the eigenvalues of this model of random matrices is a non-product Boltzmann-Gibbs measure: a Coulomb gas or one-component plasma in which the particles are confined and repell each other. This fact suggests to study more generally the concentration of measure phenomenon for Coulomb gases.

Electrostatics. Coulomb kernel in mathematical physics.

• Coulomb kernel in ${\mathbb{R}^d}$, ${d\geq2}$,

$x\in\mathbb{R}^d\mapsto g(x)= \begin{cases} \log\frac{1}{|x|} & \mbox{ if } d=2,\\ \frac{1}{|x|^{d-2}} & \mbox{ if } d\geq3 \end{cases}$

• Fundamental solution of Poisson’s equation

$\Delta g=-c_d\, \delta_0 \quad\text{where}\quad c_d = \begin{cases} 2\pi & \mbox{ if } d=2,\\ (d-2)|\mathbb{S}^{d-1}| & \mbox{ if } d\geq3. \end{cases}$

Coulomb energy and equilibrium measure.

• Coulomb energy of probability measure ${\mu}$ on ${\mathbb{R}^d}$:

$\mathcal{E}(\mu)=\iint g(x-y)\mu(\mathrm{d} x)\mu(\mathrm{d} y)\in \mathbb{R}\cup\{+\infty\}.$

• Coulomb energy with confining potential (external field)

$\mathcal{E}_V(\mu)= \mathcal{E}(\mu)+\int V(x)\mu(\mathrm{d} x).$

• Equilibrium probability measure (electrostatics)

$\mu_V=\arg\inf\mathcal{E}_V$

• If ${V}$ is stronger than log at ${\infty}$ then ${\mu_V}$ is compactly supported with density

$\frac{\Delta V}{2c_d}$

Examples of equilibrium measures

$\begin{array}{c|c|c|c} d & g & V & \mu_V \\\hline\hline 1 & 2 & \infty\mathbf{1}_{\text{interval}^c}(x) & \text{arcsine} \\ 1 & 2 & x^2 & \text{semicircle} \\ 2 & 2 & |x|^2 & \text{uniform on a disc}\\ \geq 3 & d & \Vert x\Vert^2 & \text{uniform on a ball}\\ \geq 2 & d & \text{radial} & \text{radial in a ring} \end{array}$

Coulomb gas model or one component plasma.

• Energy of ${N}$ Coulomb charges ${x_1,\ldots,x_N}$ in ${\mathbb{R}^d}$:

$\begin{array}{rcl} H_N(x_1,\ldots,x_N) &=& N\sum_{i=1}^N V(x_i) + \sum_{i\neq j} g(x_i-x_j)\\ &=&N^2\Bigr(\underbrace{\int\!V(x)\mu_N(\mathrm{d} x)+\iint_{x\neq y}g(x-y)\mu_N(\mathrm{d} x)\mu_N(\mathrm{d} y)}_{\mathcal{E}_V^{\neq}(\mu_N)}\Bigr)\\ \end{array}$

• Empirical measure:

$\mu_N=\frac{1}{N}\sum_{i=1}^N\delta_{x_i}$

• Boltzmann-Gibbs measure ${\mathbb{P}_{V,\beta}^N}$ on ${(\mathbb{R}^d)^N}$:

$\frac{\exp\left(-\frac{\beta}{2}H_N(x_1,\ldots,x_N)\right)}{Z_{V,\beta}} = \frac{\exp\left(-\frac{\beta}{2}N^2\mathcal{E}_V^{\neq}(\mu_N)\right)}{Z_{V,\beta}^N}$

• ${V}$ must be strong enough at infinity to ensure integrability
• ${\mathbb{P}_{N,\beta}}$ is neither product nor log-concave

Empirical measure and equilibrium measure

• Random empirical measure under ${\mathbb{P}_{V,\beta}^N}$:

$\mu_N = \frac{1}{N} \sum_{i=1}^N\delta_{x_i}.$

• Under mild assumptions on ${V}$, with probability one,

$\mu_N \underset{N\rightarrow\infty}{\longrightarrow} \mu_V.$

• Large Deviation Principle (Gozlan-C.-Zitt 2014)

$\frac{\log\mathbb{P}_{V,\beta}^N\Big(\,\mathrm{dist}(\mu_N,\mu_V) \ge r\Big)}{N^2} \underset{N\rightarrow\infty}{\longrightarrow} -\frac\beta2\inf_{\substack{\mathrm{dist}(\mu,\mu_V)\ge r}} \big(\mathcal{E}_V(\mu)-\mathcal{E}_V(\mu_V)\big).$

• Quantitative estimates? How to relate ${\mathrm{dist}}$ and ${\mathcal{E}_V(\cdot)-\mathcal{E}_V(\mu_V)}$?

Probability metrics and Coulomb transport inequality.

• Coulomb divergence (Large Deviations rate function)

$\mathcal{E}_V(\mu)-\mathcal{E}_V(\mu_V)$

• Coulomb metric

$\sqrt{\mathcal{E}(\mu-\nu)}$

• Bounded-Lipschitz or Fortet–Mourier distance

$\mathrm{d}_{{\mathrm{BL}}}(\mu,\nu) =\sup_{\left\|f\right\|_{\mathrm{Lip}}\leq1,\left\|f\right\|_\infty\leq1}\int f(x)(\mu-\nu)(\mathrm{d} x),$

• (Monge-Kantorovich-)Wasserstein distance

$\begin{array}{rcl} \mathrm{W}_p(\mu,\nu) &=&\inf_{\substack{(X,Y)\\X\sim\mu,Y\sim\nu}}\mathbb{E}(|X-Y|^p)^{1/p}\\ &=&\Bigr(\inf_{\pi\in\Pi(\mu,\nu)}\iint|x-y|^p\pi(\mathrm{d} x,\mathrm{d} y)\Bigr)^{1/p} \end{array}$

• Kantorovich-Rubinstein duality

$\mathrm{W}_1(\mu,\nu)=\sup_{\left\|f\right\|_{\mathrm{Lip}}\leq1}\int\!f(x)(\mu-\nu)(\mathrm{d} x).$

$\mathrm{d}_{{\mathrm{BL}}}(\mu,\nu) \leq\mathrm{W}_1(\mu,\nu) =\sup_{\left\|f\right\|_{\mathrm{Lip}}\leq1}\int\!f(x)(\mu-\nu)(\mathrm{d} x).$

Local Coulomb transport inequality.

Theorem 1 If ${D\subset~\mathbb{R}^d}$ compact, ${\mathrm{supp}(\mu+\nu)\subset D}$, ${\mathcal{E}(\mu)<\infty}$, ${\mathcal{E}(\nu)<\infty}$, then

$\mathrm{W}_1(\mu,\nu)^2\le C_D\, \mathcal{E}(\mu-\nu).$

• Constant ${C_D}$ is ${\approx\mathrm{Vol}(B_{4\mathrm{Vol}(D)})}$
• Extends Popescu local free transport inequality to any dimension ${d}$

Idea of proof:

• Potential: if ${U^\mu(x)=g*\mu(x)}$ then ${\Delta U^\mu(x)=-c_d \,\mu}$
• Electric field: ${\nabla U^\mu(x)}$. “Carré du champ”: ${|\nabla U^\mu|^2}$
• Integration by parts and Schwarz’s inequality in ${\mathbb{R}^d}$ and ${\mathrm{L}^2}$

$\begin{array}{rcl} c_d\int f(x)(\mu-\nu)(\mathrm{d} x) &=&-\int f(x)\Delta U^{\mu-\nu}(x)\mathrm{d} x\\ &\le&\left\|f\right\|_{\mathrm{Lip}}\Bigr(|D_+| \int|\nabla U^{\mu-\nu}(x)|^2\mathrm{d} x\Bigr)^{1/2} \end{array}$

• Integration by parts again

$\begin{array}{rcl} \int|\nabla U^{\mu-\nu}(x)|^2\mathrm{d} x &=&-\int U^{\mu-\nu}(x)\Delta U^{\mu-\nu}(x)\mathrm{d} x \\ &=&c_d\, \mathcal{E}(\mu-\nu). \end{array}$

• Finally ${\mathrm{W}_1(\mu,\nu)^2\leq|D_+|c_d\mathcal{E}(\mu-\nu)}$.

Coulomb transport inequality for equilibrium measures.

Theorem 2 For any probability measure ${\mu}$ on ${\mathbb{R}^d}$ with ${\mathcal{E}(\mu)<\infty}$

$\mathrm{d}_{{\mathrm{BL}}}(\mu,\mu_V)^2\le C_{{\mathrm{BL}}}\Big(\mathcal{E}_V(\mu)-\mathcal{E}_V(\mu_V)\Big).$

Moreover if ${V}$ has at least quadratic growth then

$\mathrm{W}_1(\mu,\mu_V)^2\le C_{\mathrm{W}_1}\big(\mathcal{E}_V(\mu)-\mathcal{E}_V(\mu_V)\big).$

• Free transport inequalities (${d=2}$ and ${V=+\infty}$ on ${\mathbb{R}^c}$)
• Extends results by Maïda-Maurel-Segala and Popescu to any dimension ${d}$
• Growth condition is optimal for ${\mathrm{W}_1}$

Concentration of measure for Coulomb gases

Theorem 3 If ${V}$ does has reasonable growth then for every ${\beta, N, r}$

$\mathbb{P}_{V,\beta}^N \Big(\mathrm{d}_{{\mathrm{BL}}}(\mu_N,\mu_V)\ge r \Big) \leq \mathrm{e}^{-a\beta N^2 r^2 +\mathbf 1_{d=2}(\frac{\beta}{4} N \log N) +b\beta N^{2-2/d} +c(\beta) N}.$

Moreover if ${V}$ has at least quadratic growth then ${\mathrm{W}_1}$ instead of ${\mathrm{d}_{{\mathrm{BL}}}}$.

• Large deviations principle shows that order in ${N}$ is optimal
• Explicit constants ${a,b,c}$ if ${V}$ is quadratic
• Implies Wasserstein convergence:

$\mathbb{P}_{V,\beta}^N \Big(\mathrm{W}_1(\mu_N,\mu_V)\ge r \Big)\leq \mathrm{e}^{-cN^2r^2}, \quad r\geq \begin{cases} \sqrt{\frac{\log N}{N}}& \mbox{ if } d=2,\\ N^{-1/d} & \mbox{ if } d\geq 3. \end{cases}$

• Recovers, simplifies, and extends certain results by Rougerie and Serfaty

Idea of proof:

• Starting point

$\mathbb{P}_{V,\beta}^N(\mathrm{W}_1(\mu_N,\mu_V)\geq r) =\frac{1}{Z_{V,\beta}^N}\int_{\mathrm{W}_1(\mu_N,\mu_V)\geq r}\mathrm{e}^{-\frac{\beta}{2}N^2\mathcal{E}_V^{\neq}(\mu_N)}\mathrm{d} x.$

• Normalizing constant

$\frac{1}{Z_{V,\beta}^N}\leq \exp\left\{N^2\frac{\beta}{2} \mathcal{E}_V(\mu_V) -N \left(\frac{\beta}{2}\mathcal{E}(\mu_V) -\mathrm{S}(\mu_V)\right)\right\}.$

• Regularization: ${g}$ superharmonic, ${\mu_N^{(\varepsilon)}=\mu_N*\lambda_\varepsilon}$,

$-N^2\mathcal{E}_V^{\neq}(\mu_N) \leq -N^2\mathcal{E}_V(\mu_N^{(\varepsilon)}) +N\mathcal{E}(\lambda_\varepsilon) +N\sum_{i=1}^N (V*\lambda_\varepsilon -V)(x_i).$

• Coulomb transport ${-\mathcal{E}_V(\mu_N^{(\varepsilon)})+\mathcal{E}_V(\mu_V)\leq -\frac{1}{C}\mathrm{W}_1^2(\mu_N^{(\varepsilon)},\mu_V)}$.

Concentration for spectrum of Ginibre matrices

Corollary 4 If ${M}$ is ${N\times N}$ with iid Gaussian entries of variance ${\frac{1}{N}}$ in ${\mathbb{C}}$ then

$\mathbb{P}\Big(\mathrm{W}_1(\mu_N,\mu_\bullet)\ge r \Big) \le \mathrm{e}^{ – \frac{1}{4C} N^2 r^2 +\frac{1}{2} N\log N \mathbf + N [\frac{1}{C} +\frac32 -\log\pi]}.$

• Eigenvalues of ${M}$ has distribution proportional to

$\exp(-N\sum_{i=1}^N|x_i|^2)\prod_{i<j}|x_i-x_j|^2 = \exp(-N\sum_{i=1}^N|x_i|^2-\sum_{i\neq j}g(x_i-x_j))$

• Here ${d=2}$, ${\beta=2}$, ${V=\left|\cdot\right|^2}$
• Provides ${\lim_{N\rightarrow\infty}\mathrm{W}_1(\mu_N,\mu_\bullet)=0}$ a.s.
• Open problem: prove the same for Bernoulli ${\pm 1}$ random matrices (universality)

Exponential tightness

Theorem 5 For any ${r\ge r_0}$

$\mathbb{P}_{V,\beta}^N(\mathrm{supp}(\mu_N)\not\subset B_r) = \mathbb{P}_{V,\beta}^N\bigr(\max_{1\le i\le N}|x_i|\ge r\bigr) \le\mathrm{e}^{-cNV_*(r)},$

where ${V_*(r)=\min_{|x|\geq r}V(x)}$.

• Follows by using an argument by Borot and Guionnet
• Gives that almost surely ${\varlimsup_{N\rightarrow\infty}\max_{1\le i\le N}|x_i|<\infty}$.
• Gives ${\mathrm{W}_p}$ versions of convergence and concentration

$\mathrm{W}_p^p(\mu,\nu) \leq (2M)^{p-1}\mathrm{W}_1(\mu,\nu) \leq M(2M)^{p-1}\mathrm{d}_{\mathrm{BL}}(\mu,\nu).$

For ${p=2}$ we get ${\mathbb{P}_{V,\beta}^N(\mathrm{W}_2(\mu_N,\mu_V)\ge r) \leq 2\mathrm{e}^{-cN^{3/2}r^2}}$.

Convergence in Wasserstein distance for arbitrary temperatures

Corollary 6 If ${V}$ superquadratic and ${\beta_N\geq \beta_V\frac{\log N}N}$ then under ${\mathbb{P}_{V,\beta_N}^N}$ a.s.

$\lim_{N\rightarrow\infty}\mathrm{W}_1(\mu_N,\mu_V)=0.$

Convergence at mesoscopic scale

Corollary 7 We have\ldots

• if ${d=2}$ then

$\mathbb{P}_{V,\beta}^N \Big(\mathrm{d}_{{\mathrm{BL}}}\big({\tau^{N^s}_{x_0}}\mu_N,{\tau^{N^s}_{x_0}}\mu_V\big) \geq CN^s\sqrt{\frac{\log N}{N}}\Big)\leq \mathrm{e}^{-cN\log N},$

• if ${d\geq 3}$ then

$\mathbb{P}_{V,\beta}^N \Big(\mathrm{d}_{\mathrm{BL}}\big({\tau^{N^s}_{x_0}}\mu_N,{\tau^{N^s}_{x_0}}\mu_V\big) \geq CN^{s-1/d}\Big)\leq \mathrm{e}^{-cN^{2-2/d}}.$

• If ${V}$ superquadratic then ${\mathrm{d}_{{\mathrm{BL}}}}$ can be replaced by ${\mathrm{W}_1.}$
• Note that here test functions are global, not local.

• ${\mathrm{W}_{p\geq 2}}$ versions? Popescu free transport inequalities
• Varying ${V}$ and conditional gases. ${\mathbb{P}_{V,\beta}^N(\cdot\mid x_N)=\mathbb{P}_{\widetilde V_N,\beta}^{N-1}}$ with

$\widetilde V_N=\frac{N}{N-1}V+\frac{2}{N-1}g(x_N-\cdot)$

[covered by our work since ${g}$ is superharmonic]

• Usage for CLT with GFF in all dimensions
• Weakly confining potentials and heavy-tailed ${\mu_V}$
• Universality of concentration for random matrices
• Crossover and Sanov regime ${\beta_N\propto N}$
• Concentration around the mean ${\mathbb{P}^N_{V,\beta}(\mathrm{d}(\mu_N,\mathbb{E}\mu_N)\geq r)}$ and bias ${\mathrm{d}(\mathbb{E}\mu_N,\mu_V)}$
• Dynamical aspects (in progress, see arXiv:1706.08776)