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Register a new userExact asymptotic volume and volume ratio of Schatten unit balls
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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.
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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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