Spectral properties of random Schrödinger operators with unbounded potentials

We investigate spectral properties of random Schrödinger operators H = - + _n()(1 + |n|) acting onl ²(Z ^d), where _n are independent random variables uniformly distributed on [0, 1].Research partially supported by a Sloan Doctoral Dissertation Fellowship and NSERC under grant OGP-0007901Research partially supported by NSF grant DMS-9101716     

Induced modules for vertex operator algebras

For a vertex operator algebraV and a vertex operator subalgebraV which is invariant under an automorphismg ofV of finite order, we introduce ag-twisted induction functor from the category ofg-twistedV-modules to the category ofg-twistedV-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated withV being theg-invariants in simpleV orV beingg-rational.The first author was supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.The second author was supported by NSF grant DMS-9401389.     

Random walk in random environment: A counterexample?

We describe a family of random walks in random environments which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and yet are subdiffusive in any dimensiond<.This author partially supported by NSF grant DMS 83-1080This author partially supported by NSF grant DMS-85-05020 and the Army Research Office through the Mathematical Sciences Institute at Cornell University     

Toric Self-Dual Einstein Metrics as Quotients

We use the quaternion Kähler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kähler quotients of (semi-)quaternion Kähler hyperboloids, analysing the completeness and topology, and relating them to the self-dual Einstein Hermitian metrics of Apostolov–Gauduchon and Bryant.During the preparation of this work the first and third authors were supported by NSF grant DMS-0203219. The second author was supported by the Leverhulme Trust, the William Gordon Seggie Brown Trust and an EPSRC Advanced Fellowship. The fourth author was supported by the MIUR Project Proprietà Geometriche delle Varietà Reali e Complesse. The authors are also grateful for support from EDGE, Research Training Network HPRN-CT-2000-00101, funded by the European Human Potential Programme.Acknowledgement The first author thanks the Ecole Polytechnique, Palaiseau and the Università di Roma La Sapienza for hospitality and support. The third author would like to thank the Università di Roma La Sapienza, I.N.d.A.M, M.P.I-Bonn, and IHES as parts of this paper were written during his visits there. The fourth named author would like to thank University of New Mexico for hospitality and support. The authors are grateful to Paul Gauduchon, Michael Singer and Pavel Winternitz for invaluable discussions.     

Surfaces and Peierls contours: 3-d wetting and 2-d Ising percolation

A natural model of a discrete random surface lying above a two-dimensional substrate is presented and analyzed. An identification of the level curves of the surface with the Peierls contours of Ising spin configurations leads to an exactly solvable free energy, with logarithmically divergent specific heat. The thermodynamic critical point is shown to be a wetting transition at which the surface height diverges. This is so even though the surface has no downward fingers and hence no entropic repulsion from the substrate.Dedicated to Roland DobrushinResearch supported in part by NSF Grant No. DMS-8514834 and AFOSR Contract No. F49620-86-C0130 under the U.R.I. program     

Simulated annealing via Sobolev inequalities

We use Sobolev inequalities to study the simulated annealing algorithm. This approach takes advantage of the local time reversibility of the process and yields the optimal freezing schedule as well as quantitative information about the rate at which the process is tending to its ground state.Research supported in part by NSF Grant DMS-8609944Research supported in part by NSF Grant DMS-8611487 and ARO DAAL03-86-K-0171     

Higher algebraic structures and quantization

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in ad+1 dimensional topological theory to manifolds of dimension less thand+1. We then construct a generalized path integral which ind+1 dimensions reduces to the standard one and ind dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.The author is supported by NSF grant DMS-8805684, a Presidential Young Investigators award DMS-9057144, and by the O'Donnell Foundation. He warmly thanks the Geometry Center at the University of Minnesota for their hospitality while this work was undertaken     

Exponential decay near resonance,without analyticity

We consider a quantum mechanical model which displays the behaviour associated with having a resonance or metastable state. The Hamiltonian depends on a parameter . When =0, there is an eigenstate ₀; when 0, ₀ dissolves into the continuous spectrum, showing approximate exponential decay. We prove this result without using dilatation analyticity. The model describes a two-state atom coupled to the quantized radiation field. The state space of the field is truncated, so that only the vacuum and one-photon states are included.This work was partially supported by NSF Grant DMS-8922941     

On the measure of the spectrum for the almost Mathieu operator

We obtain partial results on the conjecture that for the almost Mathieu operator at irrational frequency, , the measure of the spectrum,S(, , )=|4–2|. For ||2 we show that if _n is rational and irrational, then .Dedicated to Res Jost and Arthur WightmanResearch partially supported by U.S. NSF grant number DMS-8801918 and by BSF under grant number 88-00325     

Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics

A convergence theorem of the fractional step Lax-Friedrichs scheme and Godunov scheme for an inhomogeneous system of isentropic gas dynamics (1<5/3) is established by using the framework of compensated compactness. Meanwhile, a corresponding existence theorem of global solutions with large data containing the vacuum is obtained.Partially supported by U.S. NSF Grant # DMS-850403     

Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation

For independent translation-invariant irreducible percolation models, it is proved that the infinite cluster, when it exists, must be unique. The proof is based on the convexity (or almost convexity) and differentiability of the mean number of clusters per site, which is the percolation analogue of the free energy. The analysis applies to both site and bond models in arbitrary dimension, including long range bond percolation. In particular, uniqueness is valid at the critical point of one-dimensional 1/x–y² models in spite of the discontinuity of the percolation density there. Corollaries of uniqueness and its proof are continuity of the connectivity functions and (except possibly at the critical point) of the percolation density. Related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent ; e.g., 1+ (/2–1)/(–1).Research supported in part by NSF Grant PHY-8605164Research supported in part by the NSF through a grant to Cornell UniversityResearch supported in part by NSF Grant DMS-8514834     

The computation of characteristic classes of lattice gauge fields

AGL(p,C)-valued lattice gauge fieldu on a simplicial complex determines a principalGL(p,C)-bundle if the plaquette products are sufficiently small with respect to the maximum distortion coefficient of the transporters. A representative cocyclec _q for theq ^th Chern class of can be computed on each 2q-simplex by takingc _q() to be the intersection number of a certain singular 2q-cubeM with a Schubert-type variety _q in the space of allp×p matrices. This reduces to the solution of polynomial equations with coefficients coming fromu and thus avoids numerical integration or cooling-type procedures. An application of this method is suggested for the computation of the topological charge of anSU(3)-valued lattice gauge field on a 4-complex.Partially supported by NSF grant DMS 8607168Partially supported by PSC-CUNY and by NSF grant DMS 8805485     

Perturbative expansion of Chern-Simons theory with non-compact gauge group

Naive imitation of the usual formulas for compact gauge group in quantizing three dimensional Chern-Simons gauge theory with non-compact gauge group leads to formulas that are wrong or unilluminating. In this paper, an appropriate modification is described, which puts the perturbative expansion in a standard manifestly unitary format. The one loop contributions (which differ from naive extrapolation from the case of compact gauge group) are computed, and their topological invariance is verified.Research supported in part by NSF Grant PHY86-20266     

A note on the Ising model in high dimensions

We consider thed-dimensional Ising model with a nearest neighbor ferromagnetic interactionJ(d)=1/4d. We show that asd the+phase (and the — phase) approaches a product measure with density given by the mean field approximation. In particular the spontaneous magnetization converges to its mean field value. A similar result holds for the unique Gibbs measure of the system subject to an external fieldh0.Part of this work was done while this author was visiting Rutgers University, supported by NSF grant DMR-86-12369 and Princeton University, support by NSF grant PHY-85-15288-A01Partially supported by a NSF grant to Cornell UniversityPartially supported by NSF grant DMR 86-12369Supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell and by a NSF grant to Cornell University. This author was supported by the NSF grant DMR-86-12369 while visiting Rutgers University (when this work was started). On leave from São Paulo University     

A necessary and sufficient condition for the existence of multisolitons in a self-dual gauged sigma model

This paper presents a resolution of the gaugedO(3) sigma model proposed by B.J. Schroers in which the matter field ø mapsR ² intoS ² while the vector gauge potential gives rise to a magnetic field. It is shown that for each natural numberN there are solutions to saturate the classical energy lower boundE4N for the field configurations in the topological family deg(ø)=N if and only ifN1. Furthermore the solutions obtained depend on at least 4N–3 continuous parameters, the associated magnetic flux can assume its value in an open interval, and the decay rates of the field strengths may be specified in a suitable range. These solutions are multisolitons represented byN prescribed lumps of the magnetic field, simulatingN identical particles in equilibrium, and are governed by a nonlinear elliptic equation with both vortex and anti-vortex source terms.Research supported in part by NSF under grant DMS-9400243 (DMS-9596041).     

Open-String Vertex Algebras,Tensor Categories and Operads

We introduce notions of open-string vertex algebra, conformal open-string vertex algebra and variants of these notions. These are open-string-theoretic, noncommutative generalizations of the notions of vertex algebra and of conformal vertex algebra. Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the meromorphic center, inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center. This result gives us a general method for constructing open-string vertex algebras. Besides obvious examples obtained from associative algebras and vertex (super)algebras, we give a nontrivial example constructed from the minimal model of central charge We establish an equivalence between the associative algebras in the braided tensor category of modules for a suitable vertex operator algebra and the grading-restricted conformal open-string vertex algebras containing a vertex operator algebra isomorphic to the given vertex operator algebra. We also give a geometric and operadic formulation of the notion of grading-restricted conformal open-string vertex algebra, we prove two isomorphism theorems, and in particular, we show that such an algebra gives a projective algebra over what we call the Swiss-cheese partial operad.Acknowledgement. We would like to thank Jürgen Fuchs and Christoph Schweigert for helpful discussions and comments. We are also grateful to Jim Lepowsky for comments. The research of Y.-Z. H. is supported in part by NSF grant DMS-0070800.     

On the extremality of the disordered state for the Ising model on the Bethe lattice

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k2 is extremal if and only if is less or equal to the spin glass transition value, given by tanh( _c ^SG = 1/k.The work was partially supported by the NSF grant DMS 9504513.     

Symmetry transformations in Batalin-Vilkovisky formalism

Let us suppose that the functionalS on an odd symplectic manifold satisfies the quantum master equation e ^s = 0. We prove that in some sense every quantum observable (i.e. every functionH obeying _p (He ^s) = 0) determines a symmetry of the theory with the action functionalS. Research supported in part by NSF grant No. DMS-9201366.     

A comparison of two models of theρ-meson

A renormalization of the-propagator is presented. It is shown that if the-wave, isovector - amplitude is assumed to be dominated by this renormalized, many scattering parameters are predicted that agree well with experimental data. The model is compared with one presented by Tschang and Parkinson. It is shown that the predictions of the two models are the same, but that the renormalization model does not contain some of the theoretical problems of the Tschang and Parkinson scheme.Research partially supported by NSF Institutional Grant No. GU3220 and a grant from the Research Corporation.     

NESS theory of random steady states

A theory is presented for the properties of random steady states based upon a generalization of existing theories of nonrandom steady states. A sample calculation is presented for the energy cascade in a weakly stirred system. The theory introduces multivariable time correlation functions, and a new method for evaluating such objects is also given.Research supported by NSF grant No. CHE78-09704.Dreyfus Teacher-Scholar.