Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe maximum entropy principle and volumetric properties of Orlicz balls
226
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $M(t) = |t|^p$.
rate research
Read More
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 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.
In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and Ruschendorf and Naor and Romik unified these results by establishing a connection between $ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form [B_{phi,t}^N := Big{(s_1,ldots,s_N)inmathbb{R}^N : sum_{ i =1}^Nphi(s_i)leq t NBig},] where $phi:mathbb{R}to [0,infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-Ruschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitati
In this paper the necessary and sufficient conditions were given for Orlicz-Lorentz function space endowed with the Orlicz norm having non-squareness and local uniform non-squareness.
In this paper, for a locally compact commutative hypergroup $K$ and for a pair $(Phi_1, Phi_2)$ of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of $K,$ for the convolution $fast g$ to exist a.e., where $f$ and $g$ are arbitrary elements of Orlicz spaces $L^{Phi_1}(K)$ and $L^{Phi_2}(K)$, respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from $L^1_w(K)$ into $L^Phi_w(K)$ for a weight $w$ on a locally compact hypergroup $K$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
