What is the probability that a random integral quadratic form in nn variables has an integral zero?

We show that the density of quadratic forms in $n$ variables over $\mathbb Z_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When real quadratic forms in $n$ variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each $n$, we determine an exact expression for the probability that a random integral quadratic form in $n$ variables is isotropic (i.e., has a nontrivial zero over $\mathbb Z$), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form has an integral zero; numerically, this probability is approximately $98.3\%$.Comment: 17 pages. This article supercedes arXiv:1311.554

Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function

We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N x N random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta-function zeta(s) over sections of the critical line s=1/2+it of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function.Comment: published version with a few misprints corrected and references adde

Search for Electromagnetic Counterparts to LIGO-Virgo Candidates: Expanded Very Large Array

This paper summarizes a search for radio wavelength counterparts to candidate gravitational wave events. The identification of an electromagnetic counterpart could provide a more complete understanding of a gravitational wave event, including such characteristics as the location and the nature of the progenitor. We used the Expanded Very Large Array (EVLA) to search six galaxies which were identified as potential hosts for two candidate gravitational wave events. We summarize our procedures and discuss preliminary results.Comment: 4 pages; to appear in the New Horizons in Time Domain Astronomy, Proceedings of IAU Symposium 285, eds. R. E. M. Griffin, R. J. Hanisch & R. Seama

Liver transplant recipients’ experiences and perspectives of a telehealth-delivered lifestyle programme A qualitative study

Introduction Dietary modification and exercise are encouraged to address cardiometabolic risk factors after solid organ transplantation. However, the lived experience of attempting positive lifestyle changes for liver transplant recipients is not known. The aim of this study was to explore the experiences of liver transplant recipients and their perspectives of a 12-week telehealth lifestyle programme and assess the feasibility of this innovative health service. Methods Focus groups and one-on-one interviews were conducted with participants who had completed a 12-week, group-based, telehealth-delivered diet and exercise programme and thematic qualitative analysis was used to code and theme the data. Results In total, 19 liver transplant recipients participated in the study (25-68 years, median time since transplant 4.4 years, 63% male). Overarching themes included: (a) 'broad telehealth advantages' which highlighted that telehealth reduced the perceived burdens of face-to-face care; (b) 'impact of employment' which identified employment as a competing priority and appeared to effect involvement with the programme; (c) 'adapting Mediterranean eating pattern to meet individual needs' which identified the adaptability of the Mediterranean diet supported by sessions with the dietitian; (d) 'increasing exercise confidence' which recognised that a tailored approach facilitated confidence and acceptability of the exercise component of the programme. Discussion A telehealth lifestyle programme delivered by dietitians and exercise physiologists is an acceptable alternative to face-to-face care that can meet the needs of liver transplant recipients. There is a need to further innovate and broaden the scope of routine service delivery beyond face-to-face consultations

A new correlator in quantum spin chains

We propose a new correlator in one-dimensional quantum spin chains, the $s-$Emptiness Formation Probability ($s-$EFP). This is a natural generalization of the Emptiness Formation Probability (EFP), which is the probability that the first $n$ spins of the chain are all aligned downwards. In the $s-$EFP we let the spins in question be separated by $s$ sites. The usual EFP corresponds to the special case when $s=1$, and taking $s>1$ allows us to quantify non-local correlations. We express the $s-$EFP for the anisotropic XY model in a transverse magnetic field, a system with both critical and non-critical regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find that the magnetic field induces an interesting length scale.Comment: 6 pages, 1 figur

Random matrix theory, the exceptional Lie groups, and L-functions

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groupsComment: 14 page

Signatures of homoclinic motion in quantum chaos

Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. In this Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. By numerically studying the fluctuations of the widths of wavefunctions localized along periodic orbits we reveal the existence of an oscillatory behavior, that is explained solely in terms of the primary homoclinic motion. Furthermore, our results indicate that it survives the semiclassical limit.Comment: 5 pages, 4 figure

Europe as a multilevel federation

Peer reviewedPublisher PD

Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case

We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given for real sample covariance matrices