HighDimensional Phenomena in Convex Geometry, and Random Matrix Theory Open Access
Descriptions
 Other title
 Subject/Keyword

Convex geometry
Concentration of measure
Random matrix
 Type of item
 Thesis
 Degree grantor

University of Alberta
 Author or creator

Tikhomirov, Konstantin E
 Supervisor and department

Nicole TomczakJaegermann (Math. and Stats.)
Vlad Yaskin (Math. and Stats.)
 Examining committee member and department

Nicole TomczakJaegermann (Math. and Stats.)
Vlad Yaskin (Math. and Stats.)
Alexander Litvak (Math. and Stats.)
Arno Berger (Math. and Stats.)
Vladimir Chernousov (Math. and Stats.)
 Department

Department of Mathematical and Statistical Sciences
 Specialization

Mathematics
 Date accepted

20160616T15:42:24Z
 Graduation date

201606:Fall 2016
 Degree

Doctor of Philosophy
 Degree level

Doctoral
 Abstract

This thesis is based on six papers. The first three fall into the field of Asymptotic Geometric Analysis, the next two  Random Matrix Theory, and the sixth  highdimensional Random Walks. In the first paper, we show that for any $\varepsilon\in(0,1/2]$ and natural $n$ there is a linear subspace $E$ of $R^n$ of dimension at least $c\ln n/\ln\frac{1}{\varepsilon}$ such that $E$ is $(1+\varepsilon)$Euclidean with respect to any $1$symmetric norm in $R^n$. Here, $c>0$ is a universal constant. In the second paper, we show that, given $\varepsilon\in(0,1/2]$, a natural $n$, the space $\ell_\infty^n$, and its random subspace $E$ of dimension $m\geq 2$ uniformly distributed on the corresponding Grassmannian, $E$ is $(1+\varepsilon)$spherical with probability at least $1/2$ only if $m$ satisfies $m\leq C\varepsilon\ln n/\ln\frac{1}{\varepsilon}$ for some universal constant $C>0$. In the third paper, we show that, given an $n$dimensional convex polytope with $n+k$ vertices ($k\leq n$), its BanachMazur distance to the Euclidean ball is at least $cn/\sqrt{k}$ for some universal constant $c>0$. In the fourth paper, we prove that there are constants $c_1,c_2>0$ such that for any natural $n$ and a $2n\times n$ random matrix $A$ with i.i.d. entries $a_{ij}$ satisfying $P\{a_{ij}\lambda\leq 1\}\leq 1/2$ for all $\lambda\in R$, we have that the smallest singular value $s_{\min}(A)$ is greater than $c_1\sqrt{n}$ with probability at least $1\exp(c_2n)$. In the fifth paper, we generalize a classical theorem of Bai and Yin regarding almost sure convergence of the smallest singular values of a sequence of random matrices with i.i.d. entries. Namely, we remove the assumption that the fourth moment of the matrix entries is bounded. In the sixth paper (joint work with Pierre Youssef) we show that, given the standard $n$dimensional Brownian motion $BM_n(t)$ in $R^n$ starting at the origin, and a natural $N$, the convex hull of $BM_n(1),BM_n(2),\dots,BM_n(N)$ contains the origin with a high probability whenever $N\geq \exp(Cn)$, and contains the origin with probability close to zero whenever $N\leq\exp(cn)$. Here, $C,c>0$ are universal constants.
 Language

English
 DOI

doi:10.7939/R3SQ8QP3C
 Rights
 This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
 Citation for previous publication

K. E. Tikhomirov, Almost Euclidean sections in symmetric spaces and concentration of order statistics, J. Funct. Anal. 265 (2013), no.9, 20742088.K. E. Tikhomirov, The Randomized Dvoretzky's theorem in $\ell_\infty^n$ and the chidistribution, Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, 2116 (2014), 455463.K. E. Tikhomirov, On the distance of polytopes with few vertices to the Euclidean ball, Discrete Comput. Geom. 53 (2015), no.1, 173181.K. E. Tikhomirov. The smallest singular value of random rectangular matrices with no moment assumptions on entries. Israel Journal of Mathematics, 2016. DOI: 10.1007/s1185601612878K. Tikhomirov, The limit of the smallest singular value of random matrices with i.i.d. entries, Adv. Math. 284 (2015), 120.K. Tikhomirov and P. Youssef, When does a discretetime random walk in $R^n$ absorb the origin into its convex hull? 2015, arXiv:1410.0458, to appear in the Annals of Probability.
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