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We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschlager for classical $ell_p$-balls [Schechtman and Schmuckenschlager, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.
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The unit ball $B_p^n(mathbb{R})$ of the finite-dimensional Schatten trace class $mathcal S_p^n$ consists of all real $ntimes n$ matrices $A$ whose singular values $s_1(A),ldots,s_n(A)$ satisfy $s_1^p(A)+ldots+s_n^p(A)leq 1$, where $p>0$. Saint Raymond [Studia Math. 80, 63--75, 1984] showed that the limit $$ lim_{ntoinfty} n^{1/2 + 1/p} big(text{Vol}, B_p^n(mathbb{R})big)^{1/n^2} $$ exists in $(0,infty)$ and provided both lower and upper bounds. In this paper we determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields. A similar result is obtained for complex Schatten balls. As an application we compute the precise asymptotic volume ratio of the Schatten $p$-balls, as $ntoinfty$, thereby extending Saint Raymonds estimate in the case of the nuclear norm ($p=1$) to the full regime $1leq p leq infty$ with exact limiting behavior.
We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschlager [GAFA, Lecture notes in Math. 1469 (1991), 174--178] for $ell_p^d$-balls. As another application, we determine the precise asymptotic volume ratio for $2$-concave Orlicz spaces $ell_M^d$. Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs principle for non-interacting particles, and presents a natural approach and fresh perspective to such geometric and volumetric questions. In particular, our approach explains how the $p$-generalized Gaussian distribution occurs in problems related to the geometry of $ell_p^d$-balls, which are Orlicz balls when the Orlicz function is $M(t) = |t|^p$.
The central limit theorem for convex bodies says that with high probability the marginal of an isotropic log-concave distribution along a random direction is close to a Gaussian, with the quantitative difference determined asymptotically by the Cheeger/Poincare/KLS constant. Here we propose a generalized CLT for marginals along random directions drawn from any isotropic log-concave distribution; namely, for $x,y$ drawn independently from isotropic log-concave densities $p,q$, the random variable $langle x,yrangle$ is close to Gaussian. Our main result is that this generalized CLT is quantitatively equivalent (up to a small factor) to the KLS conjecture. Any polynomial improvement in the current KLS bound of $n^{1/4}$ in $mathbb{R}^n$ implies the generalized CLT, and vice versa. This tight connection suggests that the generalized CLT might provide insight into basic open questions in asymptotic convex geometry.
We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushimas ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yaus and Karps Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.
Thurston norms are invariants of 3-manifolds defined on their second homology vector spaces, and understanding the shape of their dual unit ball is a (widely) open problem. W. Thurston showed that every symmetric polygon in Z^2, whose vertices satisfy a parity property, is the dual unit ball of a Thurston norm on a 3-manifold. However, it is not known if the parity property on the vertices of polytopes is a sufficient condition in higher dimension or if their are polytopes, with mod 2 congruent vertices, that cannot be realized as dual unit balls of Thurston norms. In this article, we provide a family of polytopes in Z^2g that can be realized as dual unit balls of Thurston norms on 3-manifolds. These polytopes come from intersection norms on oriented closed surfaces and this article widens the bridge between these two norms.
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