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

Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies

We compute the generating functions of a O(n) model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, they were already known, and here we compute all the other topologies. We find that the generating functions (and the correlation functions of the lattice) obey the topological recursion, as usual in matrix models, i.e they are given by the symplectic invariants of their spectral curve.Comment: pdflatex, 89 pages, 12 labelled figures (15 figures at all), minor correction

Matrix eigenvalue model: Feynman graph technique for all genera

We present the diagrammatic technique for calculating the free energy of the matrix eigenvalue model (the model with arbitrary power $\beta$ by the Vandermonde determinant) to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves).Comment: Latex, 27 page

Effects of preharvest application of TH 6241 and CEPA on Vitis vinifera

Mature vines of the Vitis vinifera cultivars Thompson Seedless, Barbera, Cabernet-Sauvignon, Ruby Cabernet, and Alphonse Lavallee were treated with TH 6241 and CEPA at various concentrations. The effects at harvest time of the two growth regulators on berry weight, removal force, rupture point, total acidity, and soluble solids content were checked.Einfluß der Anwendung von TH 6241 und CEPA vor der Ernte auf Vitis viniferaReife Reben von Vitis-vinifera-Sorten (Thompson Seedless, Barbera, Cabernet Sauvignon, Ruby Cabernet, Alphonse Lavallee) wurden mit verschiedenen Konzentrationen von TH 6241 und CEPA behandelt. Die Auswirkungen der beiden Wachstumsregulatoren auf Beerengewicht, Trennkraft, Quetschfestigkeit der Beeren, Gesamtsäure und TSS wurden zum Erntezeitpunkt geprüft

All genus correlation functions for the hermitian 1-matrix model

We rewrite the loop equations of the hermitian matrix model, in a way which allows to compute all the correlation functions, to all orders in the topological $1/N^2$ expansion, as residues on an hyperelliptical curve. Those residues, can be represented diagrammaticaly as Feynmann graphs of a cubic interaction field theory on the curve.Comment: latex, 19 figure

Scattering in the adjoint sector of the c = 1 Matrix Model

Closed string tachyon emission from a traveling long string in Liouville string theory is studied. The exact collective field Hamiltonian in the adjoint sector of the c=1 matrix model is computed to capture the interaction between the tip of the long string and the closed string tachyon field. The amplitude for emission of a single tachyon quantum is obtained in a closed form using the chiral formalism.Comment: 22 pages, 2 figure

Some properties of angular integrals

We find new representations for Itzykson-Zuber like angular integrals for arbitrary beta, in particular for the orthogonal group O(n), the unitary group U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formula on n. The same methods gives also the Shatashvili's type moments. Finally we prove that, in agreement with Brezin and Hikami's observation, the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables (x_i-x_j)(y_i-y_j).Comment: 43 pages, Late

Effects of vesicular-arbuscular mycorrhizal fungi on micropropagated grapevines: Influence of endophyte strain, P fertilization and growth medium

Einfluß vesikulär-arbuskulärer Mykorrhizapilze auf in vitro vermehrte Reben: Wirkung von Pilzstamm, Phosphatdüngung und KultursubstratIn vitro vermehrte Pflanzen der Unterlagssorte Vitis berlandieri x V. rupestris 1103 P wurden in steriles Substrat verpflanzt, das mit steigenden P-Gaben angereichert war, und während der Anpassungsphase mit verschiedenen Arten vesikulär-arbuskulärer Mykorrhizapilze (VAM) beimpft. In dem Substrat ohne P-Düngung förderten alle Pilzarten das Rebenwachstum; der positive Einfluß der Beimpfung war jedoch mit zunehmender Düngungsstufe verwischt. Zwei Pilzarten waren indessen auch noch bei der höchsten P-Konzentration in der Lage, das Pflanzenwachstum zu steigern. In einem weiteren Versuch wurde der Einfluß der VAM-Art Glomus caledonicum (NICOL. et GERD.) TRAPPE et GERD. in einem Torf/Sand-Gemisch geprüft, das mit 0 oder 10 % Erde versetzt war. Obwohl der VAM-Pilz die Wurzeln in beiden Fällen besiedelte, wurde eine signifikante Wachstumssteigerung nur in dem Erde enthaltenden Substrat festgestellt

A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

We consider the two sequences of biorthogonal polynomials (p_{k,n})_k and (q_{k,n})_k related to the Hermitian two-matrix model with potentials V(x) = x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p_{n,n} as n tends to infinity. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t=0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behavior for a certain negative value of t. We also prove a general result about the interlacing of zeros of biorthogonal polynomials.Comment: 60 pages, 9 figure

Second and Third Order Observables of the Two-Matrix Model

In this paper we complement our recent result on the explicit formula for the planar limit of the free energy of the two-matrix model by computing the second and third order observables of the model in terms of canonical structures of the underlying genus g spectral curve. In particular we provide explicit formulas for any three-loop correlator of the model. Some explicit examples are worked out.Comment: 22 pages, v2 with added references and minor correction