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High-Dimensional Phenomena in Convex Geometry, and Random Matrix Theory Open Access


Other title
Convex geometry
Concentration of measure
Random matrix
Type of item
Degree grantor
University of Alberta
Author or creator
Tikhomirov, Konstantin E
Supervisor and department
Nicole Tomczak-Jaegermann (Math. and Stats.)
Vlad Yaskin (Math. and Stats.)
Examining committee member and department
Nicole Tomczak-Jaegermann (Math. and Stats.)
Vlad Yaskin (Math. and Stats.)
Alexander Litvak (Math. and Stats.)
Arno Berger (Math. and Stats.)
Vladimir Chernousov (Math. and Stats.)
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
2016-06:Fall 2016
Doctor of Philosophy
Degree level
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 - high-dimensional 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 Banach-Mazur 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.
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, 2074-2088.K. E. Tikhomirov, The Randomized Dvoretzky's theorem in $\ell_\infty^n$ and the chi-distribution, Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, 2116 (2014), 455-463.K. E. Tikhomirov, On the distance of polytopes with few vertices to the Euclidean ball, Discrete Comput. Geom. 53 (2015), no.1, 173-181.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/s11856-016-1287-8K. Tikhomirov, The limit of the smallest singular value of random matrices with i.i.d. entries, Adv. Math. 284 (2015), 1-20.K. Tikhomirov and P. Youssef, When does a discrete-time 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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