Unitary One Matrix Models: String Equations and Flows

We review the Symmetric Unitary One Matrix Models. In particular we discuss the string equation in the operator formalism, the mKdV flows and the Virasoro Constraints. We focus on the \t-function formalism for the flows and we describe its connection to the (big cell of the) Sato Grassmannian \Gr via the Plucker embedding of \Gr into a fermionic Fock space. Then the space of solutions to the string equation is an explicitly computable subspace of \Gr\times\Gr which is invariant under the flows.Comment: 20 pages (Invited talk delivered by M. J. Bowick at the Vth Regional Conference on Mathematical Physics, Edirne Turkey: December 15-22, 1991.

The Flat Phase of Crystalline Membranes

We present the results of a high-statistics Monte Carlo simulation of a phantom crystalline (fixed-connectivity) membrane with free boundary. We verify the existence of a flat phase by examining lattices of size up to $128^2$. The Hamiltonian of the model is the sum of a simple spring pair potential, with no hard-core repulsion, and bending energy. The only free parameter is the the bending rigidity $\kappa$. In-plane elastic constants are not explicitly introduced. We obtain the remarkable result that this simple model dynamically generates the elastic constants required to stabilise the flat phase. We present measurements of the size (Flory) exponent $\nu$ and the roughness exponent $\zeta$. We also determine the critical exponents $\eta$ and $\eta_u$ describing the scale dependence of the bending rigidity ($\kappa(q) \sim q^{-\eta}$) and the induced elastic constants ($\lambda(q) \sim \mu(q) \sim q^{\eta_u}$). At bending rigidity $\kappa = 1.1$, we find $\nu = 0.95(5)$ (Hausdorff dimension $d_H = 2/\nu = 2.1(1)$), $\zeta = 0.64(2)$ and $\eta_u = 0.50(1)$. These results are consistent with the scaling relation $\zeta = (2+\eta_u)/4$. The additional scaling relation $\eta = 2(1-\zeta)$ implies $\eta = 0.72(4)$. A direct measurement of $\eta$ from the power-law decay of the normal-normal correlation function yields $\eta \approx 0.6$ on the $128^2$ lattice.Comment: Latex, 31 Pages with 14 figures. Improved introduction, appendix A and discussion of numerical methods. Some references added. Revised version to appear in J. Phys.

The Factorization Method for Simulating Systems With a Complex Action

We propose a method for Monte Carlo simulations of systems with a complex action. The method has the advantages of being in principle applicable to any such system and provides a solution to the overlap problem. We apply it in random matrix theory of finite density QCD where we compare with analytic results. In this model we find non--commutativity of the limits $\mu\to 0$ and $N\to\infty$ which could be of relevance in QCD at finite density.Comment: Talk by K.N.A. at Confinement 2003, Tokyo, July 2003, 5 pages, 4 figures, ws-procs9x6.cl

The factorization method for systems with a complex action -a test in Random Matrix Theory for finite density QCD-

Monte Carlo simulations of systems with a complex action are known to be extremely difficult. A new approach to this problem based on a factorization property of distribution functions of observables has been proposed recently. The method can be applied to any system with a complex action, and it eliminates the so-called overlap problem completely. We test the new approach in a Random Matrix Theory for finite density QCD, where we are able to reproduce the exact results for the quark number density. The achieved system size is large enough to extract the thermodynamic limit. Our results provide a clear understanding of how the expected first order phase transition is induced by the imaginary part of the action.Comment: 27 pages, 25 figure

A Power Flow Method for Radial Distribution Feeders with DER Penetration

This paper presents a novel power flow method suitable for radial distribution feeders, which consists a modification of the simplified power flow concept known as the DistFlow method, already available in the literature. The proposed method relies upon a differentiated manipulation of power losses, which are taken into account in voltage calculations, unlike other simplified methods, where losses are totally neglected. As a result, calculation accuracy is greatly improved, in terms of node voltages, losses and overall active & reactive power flows. In addition, the proposed method is non-iterative and entirely linear, being easily implementable and fast in execution. The method is particularly suited for feeders with a high penetration of Distributed Energy Resources (DER), providing results that closely match those of a full non-linear power flow and are considerably more accurate than the traditional linearized distribution power flow methods, without any increase in computational burden. The new method is applied to a variety of case studies in the paper, to demonstrate its accuracy and effectiveness, comparing its performance with the simplified (linearized) DistFlow and a conventional non-linear power flow method

Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature

We present the first Monte Carlo results for supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature. The recently proposed non-lattice simulation enables us to include the effects of fermionic matrices in a transparent and reliable manner. The internal energy nicely interpolates the weak coupling behavior obtained by the high temperature expansion, and the strong coupling behavior predicted from the dual black hole geometry. The Polyakov line takes large values even at low temperature suggesting the absence of a phase transition in sharp contrast to the bosonic case. These results provide highly non-trivial evidences for the gauge/gravity duality.Comment: REVTeX4, 4 pages, 3 figure