{"id":9266,"date":"2016-12-27T16:58:17","date_gmt":"2016-12-27T15:58:17","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=9266"},"modified":"2016-12-28T12:19:37","modified_gmt":"2016-12-28T11:19:37","slug":"mind-the-gap","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2016\/12\/27\/mind-the-gap\/","title":{"rendered":"Mind the gap!"},"content":{"rendered":"

\"London<\/a><\/p>\n

Poincar\u00e9.<\/b> Recently I have spent some time thinking about the following problem: for any integer \\( {N\\geq1} \\), what is the best constant in the Poincar\u00e9 inequality<\/b> 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<\/p>\n

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

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,<\/p>\n

\\[ \\rho_N\\mathrm{Var}_{\\gamma_N}(f) \\leq\\int |\\nabla f|^2\\mathrm{d}\\gamma_N, \\]<\/p>\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} \\).<\/p>\n

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<\/b> (GUE) with density proportional to \\( {\\mathrm{e}^{-\\frac{N}{2}\\mathrm{tr}(H^2)}} \\) on \\( {N\\times N} \\) Hermitian matrices.<\/p>\n

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

\\[ E_N(x) :=\\frac{N}{2}\\sum_{i=1}^Nx_i^2+\\sum_{j<k}\\log\\frac{1}{(x_k-x_j)^2}. \\]<\/p>\n

Class of test functions.<\/b> Far beyond \\( {\\mathcal{C}^\\infty} \\) with compact support, standard approximation arguments give that the Poincar\u00e9 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)} \\).<\/p>\n

Symmetrization.<\/b> 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<\/b> 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}} \\),<\/p>\n

\\[ \\int f\\mathrm{d}\\widetilde{\\gamma}_N =\\int f_*\\mathrm{d}\\gamma_N \\]<\/p>\n

where \\( {f_*} \\) is the symmetrization of \\( {f} \\) defined by<\/p>\n

\\[ f_*(x_1,\\ldots,x_N)=\\frac{1}{N!}\\sum_{\\sigma\\in\\Sigma_N}f(x_{\\sigma(1)},\\ldots,x_{\\sigma(N)}) \\]<\/p>\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}} \\),<\/p>\n

\\[ \\int f\\mathrm{d}\\gamma_N =\\int f(x_{(1)},\\ldots,x_{(N)})\\mathrm{d}\\widetilde\\gamma_N \\]<\/p>\n

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

It is possible to compute moments of \\( {\\widetilde\\gamma_N} \\) by using the link with the GUE. For instance<\/p>\n

\\[ \\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 \\]<\/p>\n

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)} \\),<\/p>\n

\\[ \\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. \\]<\/p>\n

Since \\( {\\widetilde\\gamma_N} \\) is exchangeable it follows that<\/p>\n

\\[ \\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 \\]<\/p>\n

and<\/p>\n

\\[ \\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. \\]<\/p>\n

To compute a mixed moment \\( {x_jx_k} \\) with \\( {j\\neq k} \\), we can start from<\/p>\n

\\[ \\int(x_1+\\cdots+x_N)^2\\widetilde\\gamma_N(\\mathrm{d}x) =\\mathbb{E}(\\mathrm{tr}(H)^2)=1 \\]<\/p>\n

since \\( {\\mathrm{tr}(H)\\sim\\mathcal{N}(0,1)} \\), and use<\/p>\n

\\[ \\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) \\]<\/p>\n

to get, for any \\( {j\\neq k} \\),<\/p>\n

\\[ \\int x_jx_j\\widetilde\\gamma_N(\\mathrm{d}x) =\\int x_1x_2\\widetilde\\gamma_N(\\mathrm{d}x) =-\\frac{1}{N}. \\]<\/p>\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<\/b>.<\/p>\n

How about \\( {\\gamma_N} \\) defined on the convex set \\( {\\Lambda_N} \\)? In fact, this distribution is not isotropic<\/b>, 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,<\/p>\n

\\[ 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. \\]<\/p>\n

Nevertheless, since \\( {\\gamma_N} \\) and \\( {\\widetilde\\gamma_N} \\) agree on symmetric functions, we still have<\/p>\n

\\[ \\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, \\]<\/p>\n

and<\/p>\n

\\[ \\int(x_1^2+\\cdots+x_N^2)\\gamma_N(\\mathrm{d}x)=N. \\]<\/p>\n

Dyson.<\/b> 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<\/p>\n

\\[ \\mathrm{d}X_t=\\sqrt{2}\\mathrm{d}B_t-\\nabla E_N(X_t)\\mathrm{d}t \\]<\/p>\n

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

\\[ \\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. \\]<\/p>\n

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

Markov.<\/b> The semigroup \\( {{(P_t)}_{t\\geq0}} \\) of the process is defined by<\/p>\n

\\[ P_t(f)(x):=\\mathbb{E}(f(X_t)\\mid X_0=x) \\]<\/p>\n

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<\/b> of this semigroup is the differential operator<\/p>\n

\\[ Af=\\lim_{t\\rightarrow0^+}\\frac{P_t(f)-f}{t}=\\Delta f-\\nabla E_N\\cdot \\nabla f \\]<\/p>\n

for any smooth enough test function \\( {f} \\) (\\( {A} \\) is unbouded with a domain). The Poincar\u00e9 inequality for \\( {\\gamma_N} \\) is equivalent to an exponential decay<\/b> of the variance: \\( {\\forall f, \\forall t\\geq0} \\),<\/p>\n

\\[ \\mathrm{Var}_{\\gamma_N}(P_t(f))\\leq\\mathrm{e}^{-\\rho_N t}\\mathrm{Var}_{\\gamma_N}(f). \\]<\/p>\n

The Poincar\u00e9 inequality is also equivalent to state that \\( {A} \\) has a spectral gap<\/b>:<\/p>\n

\\[ \\mathrm{spectrum}(A)\\subset(-\\infty,-\\rho_N]\\cup\\{0\\}. \\]<\/p>\n

Convexity.<\/b> 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<\/b> (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<\/b>, it follows that the Poincar\u00e9 constant of \\( {\\gamma_N} \\) is positive:<\/p>\n

\\[ \\rho_N>0. \\]<\/p>\n

It may depend on \\( {N} \\) however. Let us compute the Hessian matrix of \\( {E_N} \\):<\/p>\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. \\]<\/p>\n

This shows in particular that for any \\( {x\\in\\mathbb{R}^N} \\), as quadratic forms,<\/p>\n

\\[ \\mathrm{Hess}(E_N)(x)\\geq NI_N. \\]<\/p>\n

Thus by the Brascamp-Lieb inequality<\/b> or by the Bakry-\u00c9mery criterion<\/b> or by the Caffarelli theorem<\/b>, it follows that the Poincar\u00e9 constant of \\( {\\gamma_N} \\) satisfies<\/p>\n

\\[ \\rho_N\\geq N. \\]<\/p>\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.<\/p>\n

The spectral gap is \\( {N} \\).<\/b> The Poincar\u00e9 inequality for \\( {\\gamma_N} \\) written for the special test function \\( {f(x)=x_1+\\cdots+x_N} \\) writes<\/p>\n

\\[ \\rho_N\\mathrm{Var}(Z_1+\\cdots+Z_N)\\leq N \\]<\/p>\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<\/p>\n

\\[ \\rho_N\\leq N. \\]<\/p>\n

Combining this fact with the previous lower bound we obtain remarkably that<\/p>\n

\\[ \\rho_N=N. \\]<\/p>\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<\/p>\n

\\[ 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). \\]<\/p>\n

This gives a conservation law<\/b>: \\( {\\sum_{i=1}^NX_{t,i}=\\sum_{i=1}^NX_{0,i}} \\) for any \\( {t\\geq0} \\).<\/p>\n

Projections.<\/b> Let us consider the hyperplane of \\( {\\mathbb{R}^N} \\)<\/p>\n

\\[ H_N:=\\{x\\in\\mathbb{R}^N:x_1+\\cdots+x_N=0\\} \\]<\/p>\n

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 It\u00f4 formula<\/b> 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}} \\).<\/p>\n

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} \\).<\/p>\n

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<\/p>\n

\\[ 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, \\]<\/p>\n

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} \\)<\/p>\n

\\[ 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, \\]<\/p>\n

which yield together finally the factorization<\/p>\n

\\[ \\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. \\]<\/p>\n

It is natural to ask about the Poincar\u00e9 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<\/b> formulated by Dyson in 1962 suggests that the Poincar\u00e9 constant for a class of sufficiently local test functions might be much larger than \\( {N} \\).<\/p>\n

Hoffman-Wielandt.<\/b> The Hoffman-Wielandt<\/a> 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<\/p>\n

\\[ \\sum_{i=1}^N(x_i(A)-x_i(B))^2\\leq\\sum_{j,k=1}^N(A_{jk}-B_{jk})^2. \\]<\/p>\n

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.<\/p>\n

Logarithmic Sobolev inequalities.<\/b> 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-\u00c9mery 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 Poincar\u00e9 inequality with twice the constant.<\/p>\n

Beyond the Gaussian case.<\/b> 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.<\/p>\n

Beyond GUE.<\/b> 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.<\/p>\n

Kannan-Lov\u00e1sz-Simonovits.<\/b> 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-Lov\u00e1sz-Simonovits (KLS) conjecture<\/b> 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 Poincar\u00e9 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 Poincar\u00e9 inequality and the fact that the Poincar\u00e9 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.<\/p>\n

Note.<\/b> This post is inspired from numerous conversations with my colleague Joseph Lehec<\/a> on Dyson Brownian motion and related topics.<\/p>\n

Further reading.<\/b><\/p>\n