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

Absolute single photoionization cross-sections of Br3+: Experiment and theory

Absolute single photoionization cross section measurements for Br3+ ions are reported in the photon energy range 44.79-59.54 eV at a photon energy resolution of 21 ±3 meV. Measurements were performed at the Advanced Light Source at Lawrence Berkeley National Laboratory using the merged-beams technique. Numerous resonance features in the experimental spectrum are assigned and their energies and quantum defect values are tabulated. The cross-section measurements are also compared with Breit-Pauli R-matrix calculations with suitable agreement over the photon energy range investigated. Analysis of the measured spectrum including Rydberg resonance series identifications produced a new emperical determination of the ionizational potential of Br3+ of 46.977 ± 0.050 eV, which is 805 meV lower than the most recently published value of 47.782 eV. This disparity between our determination and the earlier published value is similar to an 843 meV shift in the accepted ionization potential published for iso-electronic Se2+ as part of this same research program

Global spectrum fluctuations for the β\beta-Hermite and β\beta-Laguerre ensembles via matrix models

We study the global spectrum fluctuations for $\beta$-Hermite and $\beta$-Laguerre ensembles via the tridiagonal matrix models introduced in \cite{dumitriu02}, and prove that the fluctuations describe a Gaussian process on monomials. We extend our results to slightly larger classes of random matrices.Comment: 43 pages, 2 figures; typos correcte

A random matrix decimation procedure relating β=2/(r+1)\beta = 2/(r+1) to β=2(r+1)\beta = 2(r+1)

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case $r=1$ of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as $\beta$-ensembles. The inter-relations give that the joint distribution of every $(r+1)$-st eigenvalue in certain $\beta$-ensembles with $\beta = 2/(r+1)$ is equal to that of another $\beta$-ensemble with $\beta = 2(r+1)$. The proof requires generalizing a conditional probability density function due to Dixon and Anderson.Comment: 19 pages, 1 figur

Decomposition of fractional quantum Hall states: New symmetries and approximations

We provide a detailed description of a new symmetry structure of the monomial (Slater) expansion coefficients of bosonic (fermionic) fractional quantum Hall states first obtained in Ref. 1, which we now extend to spin-singlet states. We show that the Haldane-Rezayi spin-singlet state can be obtained without exact diagonalization through a differential equation method that we conjecture to be generic to other FQH model states. The symmetry rules in Ref. 1 as well as the ones we obtain for the spin singlet states allow us to build approximations of FQH states that exhibit increasing overlap with the exact state (as a function of system size). We show that these overlaps reach unity in the thermodynamic limit even though our approximation omits more than half of the Hilbert space. We show that the product rule is valid for any FQH state which can be written as an expectation value of parafermionic operators.Comment: 22 pages, 8 figure

Nonperturbative QED Processes at ELI-NP

The present paper analyses the current results and pursuits the main steps required for the design of SF-QED experiments at High-Power Laser System (HPLS) of ELI-NP in Magurele, Romania. After a brief analysis of the first experiment (E-144 SLAC), which confirmed the existence of non-linear QED interactions of the high energy electrons with the photons of a laser beam, we went on to present fundamental QED processes possible to be studied at ELI-NP in a multi-photon regime. The kinematics and characteristic parameters of the laser beam interacting with electrons were presented. In the preparation of an experiment at ELI-NP, the analysis of the kinematics and dynamics of the non-linear QED interaction processes with the physical vacuum are required. Initially, the linear QED processes and the corresponding Feynman diagrams that allow to determine the amplitude of these processes are reviewed. Based on these amplitudes, the cross sections of the processes can be obtained. For multi-photon interactions it is necessary to adapt the technique of Feynman diagrams from linear QED processes to the non-linear ones, by moving to the quantum field description with dressed Dirac-Volkov states, for particles in intense EM field. They then allow evaluation of the amplitude of the physical processes and ultimately the determination of the corresponding cross section. The SF-QED processes of multi-photon interactions with strong laser fields, can be done taking into account the characteristics of the existing facilities at ELI-NP in the context of the experimental production of electron-positron-pairs and of energetic gamma-rays. We show also some upcoming experiments similar to ours, in various stages of preparation.Comment: Presented at Bucharest University Meeting 2023 https://ssffb.fizica.unibuc.ro/SSFFB/Section.php?SectID=22

Antioxidant activity, phenolic compounds and colour of red wines treated with new fining agents

Nowadays the clarification and stabilization of red wines is generally done with conventional fining agents, like bentonite and activated coal, which pose a major challenge to environmental security and wastes management. This stimulated the use of many new techniques in order to discover alternative fining agents that don’t have negative influence on color, phenolic compounds and quality parameters. The aim of this research is to determine, how alternative fining agents, in different doses, affect antioxidant activity and colour parameters of 'Cabernet Sauvignon' red wines. Experimental material is from North-East Romania and was fined with mesoporous materials, bentonite and activated coal. Discriminant analysis classified 'Cabernet Sauvignon' wines according to the different fining agents based on total polyphenolic compounds and total antioxidant activity. Alternative fining agents, as mesoporous materials, have less impact on the colour and phenolic content of red wines in contrast to activated coal and bentonite treatments that can conduct to unsatisfying characteristics. Mesoporous materials are preferable and could be an exceptional adsorbent for polyphenolic compounds

The Anatomy of Abelian and Non-Abelian Fractional Quantum Hall States

We deduce a new set of symmetries and relations between the coefficients of the expansion of Abelian and Non-Abelian Fractional Quantum Hall (FQH) states in free (bosonic or fermionic) many-body states. Our rules allow to build an approximation of a FQH model state with an overlap increasing with growing system size (that may sometimes reach unity!) while using a fraction of the original Hilbert space. We prove these symmetries by deriving a previously unknown recursion formula for all the coefficients of the Slater expansion of the Laughlin, Read Rezayi and many other states (all Jacks multiplied by Vandermonde determinants), which completely removes the current need for diagonalization procedures.Comment: modify comment in Ref. 1

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach

We solve the loop equations of the $\beta$-ensemble model analogously to the solution found for the Hermitian matrices $\beta=1$. For \beta=1$, the solution was expressed using the algebraic spectral curve of equation$y^2=U(x)$. For arbitrary$\beta$, the spectral curve converts into a Schr\"odinger equation$((\hbar\partial)^2-U(x))\psi(x)=0$with$\hbar\propto (\sqrt\beta-1/\sqrt\beta)/N$. This paper is similar to the sister paper~I, in particular, all the main ingredients specific for the algebraic solution of the problem remain the same, but here we present the second approach to finding a solution of loop equations using sectorwise definition of resolvents. Being technically more involved, it allows defining consistently the B-cycle structure of the obtained quantum algebraic curve (a D-module of the form$y^2-U(x)$, where$[y,x]=\hbar$) and to construct explicitly the correlation functions and the corresponding symplectic invariants$F_h$, or the terms of the free energy, in 1/N^2$-expansion at arbitrary $\hbar$. The set of "flat" coordinates comprises the potential times $t_k$ and the occupation numbers \widetilde{\epsilon}_\alpha$. We define and investigate the properties of the A- and B-cycles, forms of 1st, 2nd and 3rd kind, and the Riemann bilinear identities. We use these identities to find explicitly the singular part of$\mathcal F_0$that depends exclusively on$\widetilde{\epsilon}_\alpha$.Comment: 58 pages, 7 figure