Ginsparg-Wilson Fermions in Odd Dimensions

The Ginsparg-Wilson relation, if written in a suitable form, can be used as a condition for lattice Dirac operators of massless fermions also in odd dimensions. The fermion action with such a Dirac operator is invariant under a generalized parity transformation, which reduces to the ordinary parity transformation in the (naive) continuum limit. The fermion measure, however, transforms non-trivially under the generalized parity transformation, and hence the parity anomaly arises solely from the fermion measure. The analogy to the lattice construction of chiral gauge theories in even dimensions is clarified by considering a dimensional reduction. We also propose a natural definition of a lattice Chern-Simons term, which is consistent with odd dimensional Ginsparg-Wilson fermions.Comment: 15 pages, no figures, final version published in JHE

Probabilistic Convergence Guarantees for Type II Pulse Coupled Oscillators

We show that a large class of pulse coupled oscillators converge with high probability from random initial conditions on a large class of graphs with time delays. Our analysis combines previous local convergence results, probabilistic network analysis, and a new classification scheme for Type II phase response curves to produce rigorous lower bounds for convergence probabilities based on network density. These bounds are then used to develop a simple, fast and rigorous computational analytic technique. These results suggest new methods for the analysis of pulse coupled oscillators, and provide new insights into the operation of biological Type II phase response curves and also the design of decentralized and minimal clock synchronization schemes in sensor nets.Comment: 5 pages, 3 figures, for submission to PR

Convergence of the Gaussian Expansion Method in Dimensionally Reduced Yang-Mills Integrals

We advocate a method to improve systematically the self-consistent harmonic approximation (or the Gaussian approximation), which has been employed extensively in condensed matter physics and statistical mechanics. We demonstrate the {\em convergence} of the method in a model obtained from dimensional reduction of SU($N$) Yang-Mills theory in $D$ dimensions. Explicit calculations have been carried out up to the 7th order in the large-N limit, and we do observe a clear convergence to Monte Carlo results. For $D \gtrsim 10$ the convergence is already achieved at the 3rd order, which suggests that the method is particularly useful for studying the IIB matrix model, a conjectured nonperturbative definition of type IIB superstring theory.Comment: LaTeX, 4 pages, 5 figures; title slightly changed, explanations added (16 pages, 14 figures), final version published in JHE

Relationships between log N-log S and celestial distribution of gamma-ray bursts

The apparent conflict between log N-log S curve and isotropic celestial distribution of the gamma ray bursts is discussed. A possible selection effect due to the time profile of each burst is examined. It is shown that the contradiction is due to this selection effect of the gamma ray bursts

Singular Vertices in the Strong Coupling Phase of Four-Dimensional Simplicial Gravity

We study four-dimensional simplicial gravity through numerical simulation with special attention to the existence of singular vertices, in the strong coupling phase, that are shared by abnormally large numbers of four-simplices. We attempt to cure this disease by adding a term to the action which suppresses such singular vertices. For a sufficiently large coefficient of the additional term, however, the phase transition disappears and the system is observed to be always in the branched polymer phase for any gravitational constant.Comment: 11 pages, 7 Postscript figure

On the Quantum Geometry of String Theory

The IKKT or IIB matrix model has been proposed as a non-perturbative definition of type IIB superstring theories. It has the attractive feature that space--time appears dynamically. It is possible that lower dimensional universes dominate the theory, therefore providing a dynamical solution to the reduction of space--time dimensionality. We summarize recent works that show the central role of the phase of the fermion determinant in the possible realization of such a scenario.Comment: 3 pages, 2 figures, Lattice2001(surfaces

The continuum limit of the non-commutative lambda phi^4 model

We present a numerical study of the \lambda \phi^{4} model in three Euclidean dimensions, where the two spatial coordinates are non-commutative (NC). We first show the explicit phase diagram of this model on a lattice. The ordered regime splits into a phase of uniform order and a ``striped phase''. Then we discuss the dispersion relation, which allows us to introduce a dimensionful lattice spacing. Thus we can study a double scaling limit to zero lattice spacing and infinite volume, which keeps the non-commutativity parameter constant. The dispersion relation in the disordered phase stabilizes in this limit, which represents a non-perturbative renormalization. From its shape we infer that the striped phase persists in the continuum, and we observe UV/IR mixing as a non-perturbative effect.Comment: 3 pages, 3 figures, talk presented by W.B. at the 11th Regional Conference on Mathematical Physics, Tehran, May 3-6, 200