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On the probability that a discrete complex random matrix is singular
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Title
On the probability that a discrete complex random matrix is singular
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ETD_1574
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000051423
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eng
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theses
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Mathematics
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(authority = ETD-LCSH)
Topic
Matrices
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(authority = ETD-LCSH)
Topic
Combinatorial analysis
Abstract
Let n be a large integer and M_n be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this thesis is to prove a general upper bound for the probability that M_n is singular.
For a constant 0 < p < 1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of M_n satisfy this
property, then the probability that M_n is singular is at most (p^(1/r) + o(1) )^n. All of the results in this thesis hold for any characteristic zero integral domain replacing the complex numbers.
In the special case where the entries of M_n are "fair coin flips" (taking the values +1, -1 each with probability 1/2), our general bound implies that the probability that
Mn is singular is at most (1/[square root]2 + o(1))^n, improving on the previous best upper bound of (3/4 + o(1))^n, proved by Tao and Vu in 2007.
In the special case where the entries of M_n are "lazy coin flips" (taking values +1, -1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that M_n is singular is at most (1/2 + o(1))^n, which is asymptotically sharp.
Our method is a refinement of those from Kahn, Komlos, and Szemeredi in 1995 and Tao and Vu in 2007. In particular, we make a critical use of the Structure Theorem from Tao and Vu in 2007, which was obtained using tools from additive combinatorics.
One key lemma for extending our results to the complex numbers follows from a more general result about characteristic zero integral domains. We show that any
finite system S in a characteristic zero integral domain can be mapped to Z/QZ, for infinitely many primes Q, preserving all algebraic incidences in S . This can be seen as a generalization of the well-known Freiman isomorphism lemma, which asserts that any finite subset of a torsion-free group can be mapped into Z/QZ, preserving all linear incidences.
As applications, we derive several combinatorial results (such as sum-product estimates) for a finite set in a characteristic zero integral domain. As C is a characteristic zero integral domain, this allows us to obtain new proofs for some recent results concerning finite sets of complex numbers, without relying on the topology of the plane.
For a constant 0 < p < 1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of M_n satisfy this
property, then the probability that M_n is singular is at most (p^(1/r) + o(1) )^n. All of the results in this thesis hold for any characteristic zero integral domain replacing the complex numbers.
In the special case where the entries of M_n are "fair coin flips" (taking the values +1, -1 each with probability 1/2), our general bound implies that the probability that
Mn is singular is at most (1/[square root]2 + o(1))^n, improving on the previous best upper bound of (3/4 + o(1))^n, proved by Tao and Vu in 2007.
In the special case where the entries of M_n are "lazy coin flips" (taking values +1, -1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that M_n is singular is at most (1/2 + o(1))^n, which is asymptotically sharp.
Our method is a refinement of those from Kahn, Komlos, and Szemeredi in 1995 and Tao and Vu in 2007. In particular, we make a critical use of the Structure Theorem from Tao and Vu in 2007, which was obtained using tools from additive combinatorics.
One key lemma for extending our results to the complex numbers follows from a more general result about characteristic zero integral domains. We show that any
finite system S in a characteristic zero integral domain can be mapped to Z/QZ, for infinitely many primes Q, preserving all algebraic incidences in S . This can be seen as a generalization of the well-known Freiman isomorphism lemma, which asserts that any finite subset of a torsion-free group can be mapped into Z/QZ, preserving all linear incidences.
As applications, we derive several combinatorial results (such as sum-product estimates) for a finite set in a characteristic zero integral domain. As C is a characteristic zero integral domain, this allows us to obtain new proofs for some recent results concerning finite sets of complex numbers, without relying on the topology of the plane.
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viii, 84 p. : ill.
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Ph.D.
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Includes bibliographical references (p. 80-82)
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by Philip J. Wood
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Philip J.
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Philip J. Wood
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Vu
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Van
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Van H Vu
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Kahn
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Jeff
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Jeff Kahn
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Komlos
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Janos Komlos
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Alan
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Alan Frieze
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Rutgers University
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Graduate School - New Brunswick
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2009
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2009-05
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xx
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Rutgers University Electronic Theses and Dissertations
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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doi:10.7282/T3CC10WB
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ETD doctoral
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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