Global Monte Carlo for fermions using ordered statistics

In this talk I discuss a new possibility for stochastic representation of the fe rmion determinant. The method can be used for global Monte Carlo fermion algorit hms and is tested in the case of the Schwinger model.Comment: 3 pages, Lattice2003 (Machines I

A geometric approach to free variable loop equations in discretized theories of 2D gravity

We present a self-contained analysis of theories of discrete 2D gravity coupled to matter, using geometric methods to derive equations for generating functions in terms of free (noncommuting) variables. For the class of discrete gravity theories which correspond to matrix models, our method is a generalization of the technique of Schwinger-Dyson equations and is closely related to recent work describing the master field in terms of noncommuting variables; the important differences are that we derive a single equation for the generating function using purely graphical arguments, and that the approach is applicable to a broader class of theories than those described by matrix models. Several example applications are given here, including theories of gravity coupled to a single Ising spin ($c = 1/2$), multiple Ising spins ($c = k/2$), a general class of two-matrix models which includes the Ising theory and its dual, the three-state Potts model, and a dually weighted graph model which does not admit a simple description in terms of matrix models.Comment: 40 pages, 8 figures, LaTeX; final publication versio

Two Dimensional QCD as a String Theory

I explore the possibility of finding an equivalent string representation of two dimensional QCD. I develop the large N expansion of the ${\rm QCD_2}$ partition function on an arbitrary two dimensional Euclidean manifold. If this is related to a two-dimensional string theory then many of the coefficients of the ${1\over N}$ expansion must vanish. This is shown to be true to all orders, giving strong evidence for the existence of a string representation.Comment: 24 page

c>1 Non-Critical Strings and Large-N Matrix Field Theory

Motivated by a possible relativistic string description of hadrons we use a discretised light-cone quantisation and Lanczos algorithm to investigate the phase structure of phi^3 matrix field theory in the large N limit. In 1+1 dimensions we confirm the existence of Polyakov's non-critical string theory at the boundary between parton-like and string-like phases, finding critical exponents for longitudinal oscillations equal to or consistent with those given by a mean field argument. The excitation spectrum is finite, possibly discrete. We calculate light-cone structure functions and find evidence that the probability Q(x) of a parton in the string carrying longitudinal momentum fraction between x and x+dx has support on all 0<x<1, despite the average number of partons being infinite.Comment: 9 pages LateX + 7 figures uuencode

The Kosterlitz-Thouless Phenomenon on a Fluid Random Surface

The problem of a periodic scalar field on a two-dimensional dynamical random lattice is studied with the inclusion of vortices in the action. Using a random matrix formulation, in the continuum limit for genus zero surfaces the partition function is found exactly, as a function of the chemical potential for vortices of unit winding number, at a specific radius in the plasma phase. This solution is used to describe the Kosterlitz- Thouless phenomenon in the presence of 2D quantum gravity as one passes from the ultra-violet to the infra-red.Comment: 15 pages. This version to appear in Nucl.Phys.B contains less introductory material (revised

Focusing on the Fixed Point of 4D Simplicial Gravity

Our earlier renormalization group analysis of simplicial gravity is extended. A high statistics study of the volume and coupling constant dependence of the cumulants of the node distribution is carried out. It appears that the phase transition of the theory is of first order, contrary to what is generally believed.Comment: Latex, 20 pages, 6 postscript figures, published versio

Towards a Non-Perturbative Renormalization of Euclidean Quantum Gravity

A real space renormalization group technique, based on the hierarchical baby-universe structure of a typical dynamically triangulated manifold, is used to study scaling properties of 2d and 4d lattice quantum gravity. In 4d, the $\beta$-function is defined and calculated numerically. An evidence for the existence of an ultraviolet stable fixed point of the theory is presentedComment: 12 pages Latex + 1 PS fi

Lattice Quantum Gravity: Review and Recent Developments

We review the status of different approaches to lattice quantum gravity indicating the successes and problems of each. Recent developments within the dynamical triangulation formulation are then described. Plenary talk at LATTICE 95 July 11-15, Melbourne, Australia.Comment: 12 pages, 8 figure