Philip Matchett Wood

Universality and the circular law for sparse random matrices

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

Random doubly stochastic tridiagonal matrices

1/n^{1-\alpha}

General anesthesia suppresses normal heart rate variability in humans

where

Convergence of the spectral measure of non normal matrices

0<\alpha\le1

Spectra of nearly Hermitian random matrices

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.

The Law and Practice of International Finance.

Let 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∈ n chosen uniformly at random in the set n. A simple algorithm is presented to allow direct sampling from the uniform distribution on 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

Low-Degree Factors of Random Polynomials

The human heart normally exhibits robust beat-to-beat heart rate variability (HRV). The loss of this variability is associated with pathology, including disease states such as congestive heart failure (CHF). The effect of general anesthesia on intrinsic HRV is unknown. In this prospective, observational study we enrolled 100 human subjects having elective major surgical procedures under general anesthesia. We recorded continuous heart rate data via continuous electrocardiogram before, during, and after anesthesia, and we assessed HRV of the R-R intervals. We assessed HRV using several common metrics including Detrended Fluctuation Analysis (DFA), Multifractal Analysis, and Multiscale Entropy Analysis. Each of these analyses was done in each of the four clinical phases for each study subject over the course of 24 h: Before anesthesia, during anesthesia, early recovery, and late recovery. On average, we observed a loss of variability on the aforementioned metrics that appeared to correspond to the state of general anesthesia. Following the conclusion of anesthesia, most study subjects appeared to regain their normal HRV, although this did not occur immediately. The resumption of normal HRV was especially delayed on DFA. Qualitatively, the reduction in HRV under anesthesia appears similar to the reduction in HRV observed in CHF. These observations will need to be validated in future studies, and the broader clinical implications of these observations, if any, are unknown.

Universality of the ESD for a fixed matrix plus small random noise: A stability approach

We discuss regularization by noise of the spectrum of large random nonnormal 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.

Irreducibility of Random Polynomials

We consider the eigenvalues and eigenvectors of matrices of the form M + P, where M is an n by n Wigner random matrix and P is an arbitrary n by n deterministic matrix with low rank. In general, we show that none of the eigenvalues of M + P need be real, even when P has rank one. We also show that, except for a few outlier eigenvalues, most of the eigenvalues of M + P are within 1/n of the real line, up to small order corrections. We also prove a new result quantifying the outlier eigenvalues for multiplicative perturbations of the form S ( I + P ), where S is a sample covariance matrix and I is the identity matrix. We extend our result showing all eigenvalues except the outliers are close to the real line to this case as well. As an application, we study the critical points of the characteristic polynomials of nearly Hermitian random matrices.

FIXED AND FLOATING CHARGES

Introduction to financial law. Jurisdictions of the world. Principles of world insolvency law. Bank term loans and syndicated credits. International bond issues and capital markets. Trusts in financial transactions. Set-off and netting. Payment and securities clearing systems. Security interests and title finance. Special financings: projects, acquisitions, real property, ships, aircraft. Securitisations Derivatives. Regulation of international finance. Conflict of laws and international finance. Conclusion.