From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.Comment: 41 pages, 5 figures. Submitted to Journal of Statistical Physics. Changes in this revision: recomputed Monte Carlo simulations, added reference [19], fit into margins, performed minor editin

Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited

We derive expansions of the resolvent Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we give another proof of the derivation of an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn. We conclude with a brief discussion on the derivation of the probability distribution function of the corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and Gaussian Symplectic Ensembles (GSEn)

CMB Constraints on WIMP Annihilation: Energy Absorption During the Recombination Epoch

We compute in detail the rate at which energy injected by dark matter annihilation heats and ionizes the photon-baryon plasma at z ~ 1000, and provide accurate fitting functions over the relevant redshift range for a broad array of annihilation channels and DM masses. The resulting perturbations to the ionization history can be constrained by measurements of the CMB temperature and polarization angular power spectra. We show that models which fit recently measured excesses in 10-1000 GeV electron and positron cosmic rays are already close to the 95% confidence limits from WMAP. The recently launched Planck satellite will be capable of ruling out a wide range of DM explanations for these excesses. In models of dark matter with Sommerfeld-enhanced annihilation, where sigma v rises with decreasing WIMP velocity until some saturation point, the WMAP5 constraints imply that the enhancement must be close to saturation in the neighborhood of the Earth.Comment: 17 pages, 6 figures, v2 extends discussion of constraints on Sommerfeld-enhanced model

Two ways to solve ASEP

The purpose of this article is to describe the two approaches to compute exact formulas (which are amenable to asymptotic analysis) for the probability distribution of the current of particles past a given site in the asymmetric simple exclusion process (ASEP) with step initial data. The first approach is via a variant of the coordinate Bethe ansatz and was developed in work of Tracy and Widom in 2008-2009, while the second approach is via a rigorous version of the replica trick and was developed in work of Borodin, Sasamoto and the author in 2012.Comment: 10 pages, Chapter in "Topics in percolative and disordered systems

Universal gap fluctuations in the superconductor proximity effect

Random-matrix theory is used to study the mesoscopic fluctuations of the excitation gap in a metal grain or quantum dot induced by the proximity to a superconductor. We propose that the probability distribution of the gap is a universal function in rescaled units. Our analytical prediction for the gap distribution agrees well with exact diagonalization of a model Hamiltonian

The Photophysics of the Carrier of Extended Red Emission

Interstellar dust contains a component which reveals its presence by emitting a broad, unstructured band of light in the 540 to 950 nm wavelength range, referred to as Extended Red Emission (ERE). The presence of interstellar dust and ultraviolet photons are two necessary conditions for ERE to occur. This is the basis for suggestions which attribute ERE to an interstellar dust component capable of photoluminescence. In this study, we have collected all published ERE observations with absolute-calibrated spectra for interstellar environments, where the density of ultraviolet photons can be estimated reliably. In each case, we determined the band-integrated ERE intensity, the wavelength of peak emission in the ERE band, and the efficiency with which absorbed ultraviolet photons are contributing to the ERE. The data show that radiation is not only driving the ERE, as expected for a photoluminescence process, but is modifying the ERE carrier as manifested by a systematic increase in the ERE band's peak wavelength and a general decrease in the photon conversion efficiency with increasing densities of the prevailing exciting radiation. The overall spectral characteristics of the ERE and the observed high quantum efficiency of the ERE process are currently best matched by the recently proposed silicon nanoparticle (SNP) model. Using the experimentally established fact that ionization of semiconductor nanoparticles quenches their photoluminescence, we proceeded to test the SNP model by developing a quantitative model for the excitation and ionization equilibrium of SNPs under interstellar conditions for a wide range of radiation field densities.Comment: 42 p., incl. 8 fig. Accepted for publication by Ap

Dietary Methionine Restriction Regulates Liver Protein Synthesis and Gene Expression Independently of Eukaryotic Initiation Factor 2 Phosphorylation in Mice

Background: The phosphorylation of eukaryotic initiation factor 2 (p-eIF2) during dietary amino acid insufficiency reduces protein synthesis and alters gene expression via the integrated stress response (ISR).Objective: We explored whether a Met-restricted (MR) diet activates the ISR to reduce body fat and regulate protein balance.Methods: Male and female mice aged 3-6 mo with either whole-body deletion of general control nonderepressible 2 (Gcn2) or liver-specific deletion of protein kinase R-like endoplasmic reticulum kinase (Perk) alongside wild-type or floxed control mice were fed an obesogenic diet sufficient in Met (0.86%) or an MR (0.12% Met) diet for ≤5 wk. Ala enrichment with deuterium was measured to calculate protein synthesis rates. The guanine nucleotide exchange factor activity of eIF2B was measured alongside p-eIF2 and hepatic mRNA expression levels at 2 d and 5 wk. Metabolic phenotyping was conducted at 4 wk, and body composition was measured throughout. Results were evaluated with the use of ANOVA (P < 0.05).Results: Feeding an MR diet for 2 d did not increase hepatic p-eIF2 or reduce eIF2B activity in wild-type or Gcn2-/- mice, yet many genes transcriptionally regulated by the ISR were altered in both strains in the same direction and amplitude. Feeding an MR diet for 5 wk increased p-eIF2 and reduced eIF2B activity in wild-type but not Gcn2-/- mice, yet ISR-regulated genes altered in both strains similarly. Furthermore, the MR diet reduced mixed and cytosolic but not mitochondrial protein synthesis in both the liver and skeletal muscle regardless of Gcn2 status. Despite the similarities between strains, the MR diet did not increase energy expenditure or reduce body fat in Gcn2-/- mice. Finally, feeding the MR diet to mice with Perk deleted in the liver increased hepatic p-eIF2 and altered body composition similar to floxed controls.Conclusions: Hepatic activation of the ISR resulting from an MR diet does not require p-eIF2. Gcn2 status influences body fat loss but not protein balance when Met is restricted

Muscle strengthening intervention for boys with haemophilia: Developing and evaluating a best-practice exercise programme with boys, families and health-care professionals

Background: Muscle strengthening exercises have the potential to improve outcomes for boys with haemophilia, but it is unclear what types of exercise might be of benefit. We elicited the views of health-care professionals, boys and their families to create and assess a home-based muscle strengthening programme. Objective: To design and develop a muscle strengthening programme with health-care professionals aimed at improving musculoskeletal health, and refine the intervention by engaging boys with haemophilia and their families (Study 1). Following delivery, qualitatively evaluate the feasibility and acceptability of the exercise programme with the boys and the study's physiotherapists (Study 2). Design: A person-based approach was used for planning and designing the exercise programme, and evaluating it post-delivery. The following methods were utilized: modified nominal group technique (NGT) with health-care professionals; focus group with families; exit interviews with boys; and interviews with the study's physiotherapists. Results: Themes identified to design and develop the intervention included exercises to lower limb and foot, dosage, age accommodating, location, supervision and monitoring and incentivization. Programme refinements were carried out following engagement with the boys and families who commented on: dosage, location, supervision and incentivization. Following delivery, the boys and physiotherapists commented on progression and adaptation, physiotherapist contact, goal-setting, creating routines and identifying suitable timeframes, and a repeated theme of incentivization. Conclusions: An exercise intervention was designed and refined through engagement with boys and their families. Boys and physiotherapists involved in the intervention's delivery were consulted who found the exercises to be generally acceptable with some minor refinements necessary

Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

Our interest is in the cumulative probabilities Pr(L(t) \le l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) \le l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 page

Random matrix ensembles with an effective extensive external charge

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite matrices. Here we consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter $\beta$, while for $\beta = 2$ (random matrices with unitary symmetry), the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class.Comment: LaTeX209, 17 page