Poincaré. Recently I have spent some time thinking about the following problem: for any integer ${N\geq1}$, what is the best constant in the Poincaré inequality for the probability measure ${\gamma_N}$ on the convex set ${\Lambda_N:=\{x\in\mathbb{R}^N:x_1\leq\cdots\leq x_N\}}$ with density proportional to

$x\in\Lambda_N\mapsto\mathrm{e}^{-\frac{N}{2}\sum_{i=1}^Nx_i^2}\prod_{j<k}(x_j-x_k)^2 ?$

In other words, for every ${N\geq1}$, find the best (in fact the largest) constant ${\rho_N\geq0}$ such that for every ${\mathcal{C}^\infty}$ test function ${f:\Lambda_N\rightarrow\mathbb{R}}$ with compact support,

$\rho_N\mathrm{Var}_{\gamma_N}(f) \leq\int |\nabla f|^2\mathrm{d}\gamma_N,$

where ${\mathrm{Var}_{\gamma_N}(f):=\int f^2\mathrm{d}\gamma_N-\left(\int f\mathrm{d}\gamma_N\right)^2}$ and where ${|\nabla f|^2:=(\partial_1f)^2+\cdots+(\partial_Nf)^2}$.

The reader familiar with random matrix theory knows that ${\gamma_N}$ is the law of the eigenvalues of a random matrix drawn from the Gaussian Unitary Ensemble (GUE) with density proportional to ${\mathrm{e}^{-\frac{N}{2}\mathrm{tr}(H^2)}}$ on ${N\times N}$ Hermitian matrices.

Note that ${\gamma_N}$ is a Boltzmann-Gibbs measure, with density ${Z_N^{-1}\mathrm{e}^{-E_N(x)}}$ with

$E_N(x) :=\frac{N}{2}\sum_{i=1}^Nx_i^2+\sum_{j<k}\log\frac{1}{(x_k-x_j)^2}.$

Class of test functions. Far beyond ${\mathcal{C}^\infty}$ with compact support, standard approximation arguments give that the Poincaré inequality for ${\gamma_N}$ with constant ${\rho_N}$ remains valid for any test function in the Sobolev space ${H^2(\gamma_N):=W^{1,2}(\gamma_N)}$.

Symmetrization. Let ${\widetilde{\gamma}_N}$ be the probability measure on ${\mathbb{R}^N}$ instead of ${\Lambda_N}$ with same density as ${\gamma_N}$ up to a multiplicative normalizing factor. It is exchangeable in other words invariant by permutation of the coordinates since its density is a symmetric function of the coordinates. Obviously the normalizing factor for ${\widetilde\gamma_N}$ is ${N!}$ times the one of ${\gamma_N}$. For every bounded and measurable function ${f:\mathbb{R}^N\rightarrow\mathbb{R}}$,

$\int f\mathrm{d}\widetilde{\gamma}_N =\int f_*\mathrm{d}\gamma_N$

where ${f_*}$ is the symmetrization of ${f}$ defined by

$f_*(x_1,\ldots,x_N)=\frac{1}{N!}\sum_{\sigma\in\Sigma_N}f(x_{\sigma(1)},\ldots,x_{\sigma(N)})$

where ${\Sigma_N}$ is the symmetric group of permutations of ${\{1,\ldots,N\}}$. Also ${\gamma_N=\widetilde\gamma_N}$ on symmetric functions. Conversely for any bounded and measurable ${f:\Lambda_N\rightarrow\mathbb{R}}$,

$\int f\mathrm{d}\gamma_N =\int f(x_{(1)},\ldots,x_{(N)})\mathrm{d}\widetilde\gamma_N$

where ${x_{(1)}<\cdots<x_{(N)}}$ is the reordering of ${x_1,\ldots,x_N}$.

It is possible to compute moments of ${\widetilde\gamma_N}$ by using the link with the GUE. For instance

$\int_{\mathbb{R}^N}(x_1+\cdots+x_N)\widetilde\gamma_N(\mathrm{d}x) =\mathbb{E}(\mathrm{tr}(H))=N\mathrm{H_{11}}=0$

where ${H\sim\mathrm{GUE}}$ since ${H_{11}\sim\mathcal{N}(0,N^{-1})}$, while using ${H_{12}\sim\mathcal{N}(0,(2N)^{-1}I_2)}$,

$\int_{\mathbb{R}^N}(x_1^2+\cdots+x_N^2)\widetilde\gamma_N(\mathrm{d}x) =\mathbb{E}(\mathrm{tr}(H^2)) =N\mathbb{E}(H_{11}^2)+N(N-1)\mathbb{E}(|H_{12}|^2) =N.$

Since ${\widetilde\gamma_N}$ is exchangeable it follows that

$\int x_1\widetilde\gamma_N(\mathrm{d}x)=\cdots=\int x_N\mathrm{d}\widetilde\gamma_N =\frac{1}{N}\int(x_1+\cdots+x_N)\widetilde\gamma_N(\mathrm{d}x) =0$

and

$\int x_1^2\widetilde\gamma_N(\mathrm{d}x)=\cdots=\int x_N^2\widetilde\gamma_N(\mathrm{d}x) =\frac{1}{N}\int(x_1^2+\cdots+x_N^2)\widetilde\gamma_N(\mathrm{d}x) =1.$

To compute a mixed moment ${x_jx_k}$ with ${j\neq k}$, we can start from

$\int(x_1+\cdots+x_N)^2\widetilde\gamma_N(\mathrm{d}x) =\mathbb{E}(\mathrm{tr}(H)^2)=1$

since ${\mathrm{tr}(H)\sim\mathcal{N}(0,1)}$, and use

$\int(x_1+\cdots+x_N)^2\widetilde\gamma_N(\mathrm{d}x) =N(N-1)\int x_1x_2\widetilde\gamma_N(\mathrm{d}x) +N\int\!x_1^2\widetilde\gamma_N(\mathrm{d}x)$

to get, for any ${j\neq k}$,

$\int x_jx_j\widetilde\gamma_N(\mathrm{d}x) =\int x_1x_2\widetilde\gamma_N(\mathrm{d}x) =-\frac{1}{N}.$

This shows that ${\widetilde\gamma_N}$ is asymptotically isotropic, in the sense that its mean is zero while its covariance matrix is close to the identity ${I_N}$ as ${N\rightarrow\infty}$. Note that the interior of the support of ${\widetilde\gamma_N}$, which is ${\cap_{j\neq k}\{x\in\mathbb{R}^N:x_j\neq x_k\}}$, is not connected, and this suggests that the probability measure ${\widetilde\gamma_N}$ is in some sense artificial.

How about ${\gamma_N}$ defined on the convex set ${\Lambda_N}$? In fact, this distribution is not isotropic, even when ${N\gg1}$. As a matter of fact, let us recall that if ${Z\sim\gamma_N}$ then a well known result of random matrix theory states that almost surely, regardless of the way we choose the common probability space,

$Z_1\underset{N\rightarrow\infty}{\longrightarrow}-2 \quad\mbox{and}\quad Z_N\underset{N\rightarrow\infty}{\longrightarrow}2 \quad\mbox{while}\quad \frac{1}{N}\sum_{i=1}^N\delta_{Z_i} \underset{N\rightarrow\infty}{\overset{\mathrm{weak}}{\longrightarrow}} \frac{\sqrt{4-x^2}}{2\pi}\mathrm{1}_{[-2,2]}(x)\mathrm{d}x.$

Nevertheless, since ${\gamma_N}$ and ${\widetilde\gamma_N}$ agree on symmetric functions, we still have

$\int(x_1+\cdots+x_N)\gamma_N(\mathrm{d}x)=0,\quad \int(x_1+\cdots+x_N)^2\gamma_N(\mathrm{d}x)=1,$

and

$\int(x_1^2+\cdots+x_N^2)\gamma_N(\mathrm{d}x)=N.$

Dyson. The probability measure ${\gamma_N}$ is invariant for the irreducible Markov diffusion process ${{(X_t)}_{t\geq0}}$ on ${\Lambda_N}$ solution of the stochastic differential equation

$\mathrm{d}X_t=\sqrt{2}\mathrm{d}B_t-\nabla E_N(X_t)\mathrm{d}t$

where ${{(B_t)}_{t\geq0}}$ is a standard Brownian motion on ${\mathbb{R}^N}$, in other words

$\mathrm{d}X_{t,i} =\sqrt{2}\mathrm{d}B_{t,i} -NX_{t,i}\mathrm{d}t -2\sum_{j\neq i}\frac{1}{X_{t,j}-X_{t,i}}\mathrm{d}t, 1\leq i\leq N.$

We may call it the Dyson Ornstein-Uhlenbeck process. The equation above is a system of stochastic differential equations of interacting particles, which can be seen (via the empirical measure) as a mean-field linear approximation of a McKean-Vlasov semilinear evolution equation with singular interaction, but this is another story.

Markov. The semigroup ${{(P_t)}_{t\geq0}}$ of the process is defined by

$P_t(f)(x):=\mathbb{E}(f(X_t)\mid X_0=x)$

for any ${t\geq0}$, any bounded measurable ${f:\mathbb{R}^N\rightarrow\mathbb{R}}$, and any ${x\in\mathbb{R}^N}$. For any ${t\geq0}$ the linear operator ${P_t}$ is a contraction of ${\mathrm{L}^p(\gamma_N)}$ for any ${p\in[1,\infty]}$. In ${\mathrm{L}^2(\gamma_N)}$, the infinitesimal generator of this semigroup is the differential operator

$Af=\lim_{t\rightarrow0^+}\frac{P_t(f)-f}{t}=\Delta f-\nabla E_N\cdot \nabla f$

for any smooth enough test function ${f}$ (${A}$ is unbouded with a domain). The Poincaré inequality for ${\gamma_N}$ is equivalent to an exponential decay of the variance: ${\forall f, \forall t\geq0}$,

$\mathrm{Var}_{\gamma_N}(P_t(f))\leq\mathrm{e}^{-\rho_N t}\mathrm{Var}_{\gamma_N}(f).$

The Poincaré inequality is also equivalent to state that ${A}$ has a spectral gap:

$\mathrm{spectrum}(A)\subset(-\infty,-\rho_N]\cup\{0\}.$

Convexity. On the convex set ${\Lambda_N}$, the energy ${E_N}$ of the Boltzmann-Gibbs measure ${\gamma_N}$ is convex, and therefore ${\gamma_N}$ is log-concave (but is not isotropic). Note that in contrast ${\widetilde\gamma_N}$ is not log-concave (but is almost isotropic as ${N\rightarrow\infty}$). Thanks to a result by Bobkov, it follows that the Poincaré constant of ${\gamma_N}$ is positive:

$\rho_N>0.$

It may depend on ${N}$ however. Let us compute the Hessian matrix of ${E_N}$:

$\partial^2_{i,i}E_N(x) =N+2(N-1)\sum_{k\neq i}\frac{1}{(x_i-x_k)^2} \quad\mbox{and}\quad \partial^2_{j,k}E_N(x) =-\frac{2(N-1)}{(x_j-x_k)^2},\quad j\neq k.$

This shows in particular that for any ${x\in\mathbb{R}^N}$, as quadratic forms,

$\mathrm{Hess}(E_N)(x)\geq NI_N.$

Thus by the Brascamp-Lieb inequality or by the Bakry-Émery criterion or by the Caffarelli theorem, it follows that the Poincaré constant of ${\gamma_N}$ satisfies

$\rho_N\geq N.$

This bound comes from the strong convexity of the confinement term ${N\sum_{i=1}^Nx_i^2}$ and the convexity of the interaction term ${\sum_{j<k}\log\frac{1}{(x_j-x_k)^2}}$. The interaction term is strongly convex when ${x}$ in away from infinity, but this is not seen by global criteria.

The spectral gap is ${N}$. The Poincaré inequality for ${\gamma_N}$ written for the special test function ${f(x)=x_1+\cdots+x_N}$ writes

$\rho_N\mathrm{Var}(Z_1+\cdots+Z_N)\leq N$

where ${Z\sim\gamma_N}$. But if ${H\sim\mathrm{GUE}}$ then ${Z_1+\cdots+Z_N=\mathrm{tr}(H)\sim\mathcal{N}(0,1)}$ and thus

$\rho_N\leq N.$

Combining this fact with the previous lower bound we obtain remarkably that

$\rho_N=N.$

Actually, it turns out that the function ${f(x)=x_1+\cdots+x_N}$ is an eigenfunction of the infinitesimal generator ${A}$ associated to the spectral value ${N}$, indeed

$Af(x) =\sum_{i=1}^N\partial_iE_N(x) =\sum_{i=1}^N\left(Nx_i+2\sum_{j\neq i}\frac{1}{x_j-x_i}\right) =Nf(x).$

This gives a conservation law: ${\sum_{i=1}^NX_{t,i}=\sum_{i=1}^NX_{0,i}}$ for any ${t\geq0}$.

Projections. Let us consider the hyperplane of ${\mathbb{R}^N}$

$H_N:=\{x\in\mathbb{R}^N:x_1+\cdots+x_N=0\}$

orthogonal to ${(1,\ldots,1)}$. Let ${\pi_N}$ and ${\pi_N^\perp}$ be the orthogonal projections on ${H_N}$ and ${H_N^\perp=\mathbb{R}(1,\ldots,1)}$. The Itô formula shows that the projected processes ${{(\pi_N(X_t))}_{t\geq0}}$ and ${{(\pi_N^\perp(X_t))}_{t\geq0}}$ are independent, and that moreover the first one is a Markov diffusion process on ${H_N}$ while the second is a Brownian motion on ${H_N^\perp\equiv\mathbb{R}}$.

Similarly if ${Z\sim\gamma_N}$ then ${\pi_N(Z)}$ and ${\pi_N^\perp(Z)}$ are independent, the first one follows a Boltzmann-Gibbs measure with same density as ${\gamma_N}$ but on ${H_N}$, while the second follows a Gaussian distribution of dimension ${1}$.

The quadratic term in the exponential and the Vandermonde determinant which both appear in the density of ${\gamma_N}$ play an essential role in these projection properties. More precisely, the Pythagoras theorem gives

$x_1^2+\cdots+x_N^2 =|x|^2 =|x-\pi_N(x)|^2+|\pi_N(x)|^2 =|\pi_N^\perp(x)|^2+|\pi_N(x)|^2,$

while the formulas ${\pi_N^\perp(x)=\frac{x_1+\cdots+x_N}{N}(1,\ldots,1)}$ and ${\pi_N(x)=x-\pi_N^\perp(x)}$ give for ${j<k}$

$x_j-x_k =x_j-\frac{x_1+\cdots+x_N}{N}-\left(x_k-\frac{x_1+\cdots+x_N}{N}\right) =\pi_N(x)_j-\pi_N(x)_k,$

which yield together finally the factorization

$\mathrm{e}^{-\frac{N}{2}\sum_{i=1}^Nx_i^2}\prod_{j<k}(x_j-x_k)^2 =\mathrm{e}^{-\frac{N}{2}|\pi_N^\perp(x)|^2}\times \mathrm{e}^{-\frac{N}{2}|\pi_N(x)|^2}\prod_{j<k}(\pi_N(x)_j-\pi_N(x)_k)^2.$

It is natural to ask about the Poincaré constant for the projected process. This constant is necessarily ${\geq N}$ and one can check on quadratic functions that this bound is in fact sharp, therefore the spectral gap is not improved by this projection and remains equal to ${N}$. However, the Dyson conjecture formulated by Dyson in 1962 suggests that the Poincaré constant for a class of sufficiently local test functions might be much larger than ${N}$.

Hoffman-Wielandt. The Hoffman-Wielandt theorem for Hermitian matrices states that for any ${N\geq1}$ if ${A}$ and ${B}$ are two ${N\times N}$ Hermitian matrices with eigenvalues ${x_1(A)\leq\cdots\leq x_N(A)}$ and ${x_1(B)\leq\cdots\leq x_N(B)}$ then

$\sum_{i=1}^N(x_i(A)-x_i(B))^2\leq\sum_{j,k=1}^N(A_{jk}-B_{jk})^2.$

This shows that the ordered vector of eigenvalues is a Lipschitz function the matrix entries, and this explains why the Gaussian nature of the GUE induces nice properties for ${\gamma_N}$. This can also be seen as an alternative to the Caffarelli theorem.

Logarithmic Sobolev inequalities. Thanks to the Hoffman-Wielandt inequality, the Gaussian nature of the GUE should induces a sub-Gaussian nature for ${\gamma_N}$. Indeed again the Bakry-Émery criterion or the Caffarelli theorem give that ${\gamma_N}$ satisfies to a logarithmic Sobolev inequality with constant ${\geq N/2}$, and thus ${=N/2}$ since it implies the Poincaré inequality with twice the constant.

Beyond the Gaussian case. Many properties considered above remain valid if one replaces, in the exponential in the density of ${\gamma_N}$, the quadratic term ${x_i^2}$ by ${V(x_i)}$ where ${V:\mathbb{R}\rightarrow\mathbb{R}}$ is convex with ${\inf_{\mathbb{R}}V”>0}$. However certain rigid properties are lost, such as the fact that ${x_1+\cdots+x_N}$ is an eigenvector of ${A}$, and in particular the projected process is no longer necessarily a Markov process.

Beyond GUE. Certain aspects remain valid if one replaces, in the density of ${\gamma_N}$, the term ${(x_j-x_k)^2}$ by ${|x_j-x_k|^\beta}$ for some ${\beta>0}$. The cases ${\beta=1}$ and ${\beta=4}$ correspond respectively to the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Simplectic Ensemble (GSE). The condition ${\beta\geq1}$ seems to be necessary in order to ensure that the diffusion process ${{(X_t)}_{t\geq0}}$ does not explode in finite time.

Kannan-Lovász-Simonovits. Log-concave probability measures play a central role in the analysis and geometry of convex bodies, as being functional generalization of uniform distributions on convex bodies. In this context, the Kannan-Lovász-Simonovits (KLS) conjecture formulated in 1995 states that there exists a universal constant ${\rho\in(0,\infty)}$ such that for every ${N\geq1}$ and every log-concave probability measure ${\mu}$ on ${\mathbb{R}^N}$ with mean ${0}$ and covariance matrix ${I_N}$ (isotropy), the Poincaré constant of ${\mu}$ is larger than or equal to ${\rho}$. The feature here is the uniformity of the bound in particular with respect to the dimension ${N}$. Several mathematicians have tried to prove the conjecture. The best bound for now is ${N^{-1/4}}$. Possible counter examples should be searched among non-product probability measures due to the stability by tensor product of the Poincaré inequality and the fact that the Poincaré constant of a one dimensional log-concave distribution is controlled by its second moment. The KLS conjecture is related to other important conjectures in geometric functional analysis such that the hyperplane and the thin-shell conjectures.

Note. This post is inspired from numerous conversations with my colleague Joseph Lehec on Dyson Brownian motion and related topics.