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 paper 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 paper 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 M n is singular is at most ( 1 √ 2 + o(1)) n , improving on the previous best upper bound of ( 3 4 + o(1)) n , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. 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 [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223-240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628], which was obtained using tools from additive combinatorics.

The universality phenomenon asserts that the distribution of the eigenvalues of random matrix with i.i.d. zero mean, unit variance entries does not depend on the underlying structure of the random entries. For example, a plot of the eigenvalues of a random sign matrix, where each entry is +1 or -1 with equal probability, looks the same as an analogous plot of the eigenvalues of a random matrix where each entry is complex Gaussian with zero mean and unit variance. In the current paper, we prove a universality result for sparse random n by n matrices where each entry is nonzero with probability $1/n^{1-\alpha}$ where $0<\alpha\le1$ is any constant. One consequence of the sparse universality principle is that the circular law holds for sparse random matrices so long as the entries have zero mean and unit variance, which is the most general result for sparse random matrices to date.Comment: Published in at http://dx.doi.org/10.1214/11-AAP789 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We discuss regularization by noise of the spectrum of large random non-Normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in * -moments to a regular element a, by the addition of a polynomially vanishing Gaussian Ginibre matrix, forces the empirical measure of eigenvalues to converge to the Brown measure of a. * UMPA, CNRS UMR 5669, ENS Lyon, 46 allée d'Italie,

Let \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally in probability problems as birth and death chains with a uniform stationary distribution. We study ‘typical’ matrices T∈ \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n chosen uniformly at random in the set \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n. A simple algorithm is presented to allow direct sampling from the uniform distribution on \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n. Using this algorithm, the elements above the diagonal in T are shown to form a Markov chain. For large n, the limiting Markov chain is reversible and explicitly diagonalizable with transformed Jacobi polynomials as eigenfunctions. These results are used to study the limiting behavior of such typical birth and death chains, including their eigenvalues and mixing times. The results on a uniform random tridiagonal doubly stochastic matrices are related to the distribution of alternating permutations chosen uniformly at random.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 403–437, 2013

Objective-Terodiline, an antimuscarinic and calcium antagonist drug, was used to treat detrusor instability but was withdrawn in 1991 after provoking serious ventricular arrhythmias associated with increases in the corrected QT interval (QTc). This research was performed to relate drug induced electrocardiographic changes in asymptomatic recipients to plasma concentrations of the R( +) and S(-) terodiline enantiomers. (443 (33) and 42 (17) ms'12, paired t tests, P < 0-002 and P < 0 01 respectively) in the 12 patients in sinus rhythm. The mean (95% confidence interval) drug induced increases were 48 (23 to 74) Ms112 for QTc and 42 (13 to 70) ms'12 for QTd. These increases correlated with total plasma terodiline (QTc: r = 0-77, P < 0-006, QTd: r = 0*68, P < 0.025) and with plasma concentrations of both terodiline enantiomers.Conclusions-Terodiline increases QTc and QTd in a concentration dependent manner. It is not clear whether this is a stereoselective effect and, if so, which enantiomer is responsible. The results suggest that drug induced torsade de pointes is a type A (concentration dependent) adverse drug reaction. (Br HeartJ7 1995;74:53-56)