The scaling limit of the critical one-dimensional random Schrodinger operator

We consider two models of one-dimensional discrete random Schrodinger operators (H_n \psi)_l ={\psi}_{l-1}+{\psi}_{l +1}+v_l {\psi}_l, {\psi}_0={\psi}_{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and v_k=\sigma {\omega}_k/ \sqrt{k}. Here {\omega}_k are independent random variables with mean 0 and variance 1. We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem. In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the beta-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.Comment: 36 pages, 2 figure

The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(\beta = 1)$, complex ($\beta = 2)$ and real quaternion $(\beta = 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $\beta > 0$. In the Gaussian case a further construction valid for $\beta > 0$ is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. The resulting functional forms are shown to be a special cases of duality formulas due to Desrosiers. New derivations of the general case of Desrosiers' dualities are given. A soft edge scaling limit of the averaged characteristic polynomial is identified, and an explicit evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page

Bridge Decomposition of Restriction Measures

Motivated by Kesten's bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of SLE(8/3), and as a result can be generalized to the wider class of restriction measures. Specifically we show that the restriction hulls with index less than one can be decomposed into a Poisson Point Process of irreducible bridges in a way that is similar to Ito's excursion decomposition of a Brownian motion according to its zeros.Comment: 24 pages, 2 figures. Final version incorporates minor revisions suggested by the referee, to appear in Jour. Stat. Phy

Spectral density asymptotics for Gaussian and Laguerre β\beta-ensembles in the exponentially small region

The first two terms in the large $N$ asymptotic expansion of the $\beta$ moment of the characteristic polynomial for the Gaussian and Laguerre $\beta$-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms $o(1)$ . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.Comment: 19 page

Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process

Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover, the set of absolute values of the zeros of f has the same distribution as the set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1]. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant.Comment: 37 pages, 2 figures, updated proof

Diclofenac Prolongs Repolarization in Ventricular Muscle with Impaired Repolarization Reserve

Background: The aim of the present work was to characterize the electrophysiological effects of the non-steroidal anti- inflammatory drug diclofenac and to study the possible proarrhythmic potency of the drug in ventricular muscle. Methods: Ion currents were recorded using voltage clamp technique in canine single ventricular cells and action potentials were obtained from canine ventricular preparations using microelectrodes. The proarrhythmic potency of the drug was investigated in an anaesthetized rabbit proarrhythmia model. Results: Action potentials were slightly lengthened in ventricular muscle but were shortened in Purkinje fibers by diclofenac (20 mM). The maximum upstroke velocity was decreased in both preparations. Larger repolarization prolongation was observed when repolarization reserve was impaired by previous BaCl 2 application. Diclofenac (3 mg/kg) did not prolong while dofetilide (25 mg/kg) significantly lengthened the QT c interval in anaesthetized rabbits. The addition of diclofenac following reduction of repolarization reserve by dofetilide further prolonged QT c . Diclofenac alone did not induce Torsades de Pointes ventricular tachycardia (TdP) while TdP incidence following dofetilide was 20%. However, the combination of diclofenac and dofetilide significantly increased TdP incidence (62%). In single ventricular cells diclofenac (30 mM) decreased the amplitude of rapid (I Kr ) and slow (I Ks ) delayed rectifier currents thereby attenuating repolarization reserve. L-type calcium current (I Ca ) was slightly diminished, but the transient outward (I to ) and inward rectifier (I K1 ) potassium currents were not influenced. Conclusions: Diclofenac at therapeutic concentrations and even at high dose does not prolong repolarization markedly and does not increase the risk of arrhythmia in normal heart. However, high dose diclofenac treatment may lengthen repolarization and enhance proarrhythmic risk in hearts with reduced repolarization reserve

A systematic review of the evidence on decoupling of GDP, resource use and GHG emissions, part I: bibliometric and conceptual mapping

As long as economic growth is a major political goal, decoupling growth from resource use and emissions is a prerequisite for a sustainable net-zero emissions future. However, empirical evidence for absolute decoupling, i.e., decreasing resource use and emissions at the required scale despite continued economic growth, is scarce and scattered across different research streams. In this two-part systematic review, we assess how and to what extent decoupling has been observed and what can be learnt for addressing the sustainability and climate crisis. Based on a transparent approach, we systematically identify and screen more than 11,500 scientific papers, eventually analyzing full texts of 835 empirical studies on the relationship between economic growth (GDP), resource use (materials and energy) and greenhouse gas emissions. Part I of the review examines how decoupling has been investigated across three research streams: energy, materials and energy, and emissions. Part II synthesizes the empirical evidence and policy implications (Haberl et al. part II, in review). In part I, we examine the topical, temporal and geographical scopes, methods of analysis, institutional networks and prevalent conceptual angles. We find that in this rapidly growing literature, the vast majority of studies – decomposition, 'causality' and Environmental Kuznets Curve analysis – approach the topic from a statistical-econometric point of view, while hardly acknowledging thermodynamic principles on the role of energy and materials for socio-economic activities. A potentially fundamental incompatibility between economic growth and systemic societal changes to address the climate crisis is rarely considered. We conclude that the existing wealth of empirical evidence merits braver conceptual advances than we have seen thus far. Future work should focus on comprehensive multi-indicator long-term analyses, conceptually grounded on the fundamental biophysical basis of socio-economic activities, incorporating the role of global supply chains as well as the wider societal role and preconditions of economic growth

A systematic review of the evidence on decoupling of GDP, resource use and GHG emissions, part II: synthesizing the insights

Strategies toward ambitious climate targets usually rely on the concept of "decoupling"; that is, they aim at promoting economic growth while reducing the use of natural resources and GHG emissions. GDP growth coinciding with absolute reductions in emissions or resource use is denoted as "absolute decoupling", as opposed to "relative decoupling", where resource use or emissions increase less so than does GDP. Based on the bibliometric mapping in part I (Wiedenhofer et al., this issue), we synthesize the evidence emerging from the selected 835 peer-reviewed articles. We evaluate empirical studies of decoupling related to final/useful energy, exergy, use of material resources, as well as CO2 and total GHG emissions. We find that relative decoupling is frequent for material use as well as GHG and CO2 emissions but not for useful exergy, a quality-based measure of energy use. Primary energy can be decoupled from GDP largely to the extent to which the conversion of primary energy to useful exergy is improved. Examples of absolute long-term decoupling are rare, but recently some industrialized countries have decoupled GDP from both production- and, weaklier, consumption-based CO2 emissions. We analyze policies or strategies in the decoupling literature by classifying them into three groups: (1) Green growth, if sufficient reductions of resource use or emissions were deemed possible without altering the growth trajectory. (2) Degrowth, if reductions of resource use or emissions were given priority over GDP growth. (3) Others, e.g. if the role of energy for GDP growth was analyzed without reference to climate change mitigation. We conclude that large rapid absolute reductions of resource use and GHG emissions cannot be achieved through observed decoupling rates, hence decoupling needs to be complemented by sufficiency-oriented strategies and strict enforcement of absolute reduction targets. More research is needed on interdependencies between wellbeing, resources and emissions