G. Halász

Bio: G. Halász is an academic researcher. The author has contributed to research in topics: Probability-generating function & Characteristic function (probability theory). The author has an hindex of 1, co-authored 1 publications receiving 174 citations.

Topics in Random Matrix Theory

Abstract: The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.

Smallest singular value of a random rectangular matrix

Abstract: We prove an optimal estimate of the smallest singular value of a random sub- Gaussian matrix, valid for all dimensions. For an Nn matrix A with inde- pendent and identically distributed sub-Gaussian entries, the smallest singular value of A is at least of the order p Np n � 1 with high probability. A sharp

The Littlewood-Offord Problem and invertibility of random matrices

Abstract: We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables X_k and real numbers a_k, determine the probability P that the sum of a_k X_k lies near some number v. For arbitrary coefficients a_k of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p.

Inverse Littlewood-Offord theorems and the condition number of random discrete matrices

Abstract: Consider a random sum r)\V\ + • • • + r]nvn, where 771, . . . , rin are independently and identically distributed (i.i.d.) random signs and vi, . . . , vn are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as P(r?i^iH In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman's inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v\ , . . . , vn are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the