Asymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals

In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian matrices the probability that an interval of length $s$ contains no eigenvalues is the Fredholm determinant of the sine kernel $\sin(x-y)\over\pi(x-y)$ over this interval. A formal asymptotic expansion for the determinant as $s$ tends to infinity was obtained by Dyson. In this paper we replace a single interval of length $s$ by $sJ$ where $J$ is a union of $m$ intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect to $s$ of the determinant equals a constant (expressible in terms of hyperelliptic integrals) times $s$, plus a bounded oscillatory function of $s$ (zero of $m=1$, periodic if $m=2$, and in general expressible in terms of the solution of a Jacobi inversion problem), plus $o(1)$. Also determined are the asymptotics of the trace of the resolvent operator, which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none. The proofs use ideas from orthogonal polynomial theory.Comment: 24 page

First-principles extrapolation method for accurate CO adsorption energies on metal surfaces

We show that a simple first-principles correction based on the difference between the singlet-triplet CO excitation energy values obtained by DFT and high-level quantum chemistry methods yields accurate CO adsorption properties on a variety of metal surfaces. We demonstrate a linear relationship between the CO adsorption energy and the CO singlet-triplet splitting, similar to the linear dependence of CO adsorption energy on the energy of the CO 2$\pi$* orbital found recently {[Kresse {\em et al.}, Physical Review B {\bf 68}, 073401 (2003)]}. Converged DFT calculations underestimate the CO singlet-triplet excitation energy $\Delta E_{\rm S-T}$, whereas coupled-cluster and CI calculations reproduce the experimental $\Delta E_{\rm S-T}$. The dependence of $E_{\rm chem}$ on $\Delta E_{\rm S-T}$ is used to extrapolate $E_{\rm chem}$ for the top, bridge and hollow sites for the (100) and (111) surfaces of Pt, Rh, Pd and Cu to the values that correspond to the coupled-cluster and CI $\Delta E_{\rm S-T}$ value. The correction reproduces experimental adsorption site preference for all cases and obtains $E_{\rm chem}$ in excellent agreement with experimental results.Comment: Table sent as table1.eps. 3 figure

Integrable structure of Ginibre's ensemble of real random matrices and a Pfaffian integration theorem

In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005)], an exact solution was reported for the probability p_{n,k} to find exactly k real eigenvalues in the spectrum of an nxn real asymmetric matrix drawn at random from Ginibre's Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined

Integrated analysis of environmental and genetic influences on cord blood DNA methylation in new-borns

Epigenetic processes, including DNA methylation (DNAm), are among the mechanisms allowing integration of genetic and environmental factors to shape cellular function. While many studies have investigated either environmental or genetic contributions to DNAm, few have assessed their integrated effects. Here we examine the relative contributions of prenatal environmental factors and genotype on DNA methylation in neonatal blood at variably methylated regions (VMRs) in 4 independent cohorts (overall n = 2365). We use Akaike’s information criterion to test which factors best explain variability of methylation in the cohort-specific VMRs: several prenatal environmental factors (E), genotypes in cis (G), or their additive (G + E) or interaction (GxE) effects. Genetic and environmental factors in combination best explain DNAm at the majority of VMRs. The CpGs best explained by either G, G + E or GxE are functionally distinct. The enrichment of genetic variants from GxE models in GWAS for complex disorders supports their importance for disease risk