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     

Absence of continuous symmetry breaking in a one-dimensional n−2 model

For a one-dimensional array ofS ^N–1 spins (N 2) with isotropic pair interactions (and more general systems) with J(j–i) obeying sup_n[n^–1 ₁ ⁿ j ²|J(j)|]<, we prove that every equilibrium state is invariant under the natural action ofSO(N). In particular, there is no long-range order of the conventional type. Included is the caseJ(n)=n ^–2.Research partially supported by U.S.N.S.F. Grant No. MCS-78-01885.S. Fairchild Scholar at Caltech. On leave from Departments of Mathematics and Physics, Princeton University, Princeton, New Jersey 08544.     

Rigorous proof of the high-temperature Josephson inequality for critical exponents

I give a rigorous proof of the high-temperature Josephson inequalitydv 2–, following the original ideas of Josephson. The proof is applicable to a class of models including the ferromagnetic Ising model and the ⁴ lattice field theory.Research supported in part by NSF Grant PHY 78-23952.     

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A theorem on ergodicity of two-dimensional hyperbolic billiards

Ergodicity of two-dimensional billiards which satisfy some general conditions is proved. This theorem is applied to one concrete class of billiards that contains, in particular, billiards in the stadium.This paper was completed during the author's stay at Rutgers University where he was supported by NSF Grant DHS 89-01545 and at Courant Institute of Mathematical Sciences where he was supported by a Grant from the Alfred P. Sloan Foundation     

On the second eigenfunctions of the Laplacian in R2

A conjecture about the nodal line of a second eigenfunction states that the nodal line of a second eigenfunction divides the domain by intersecting with the boundary of transversely, where is a bounded convex domain ofR ². We prove this conjecture provided has a symmetry. Also, we prove the multiplicity of the second eigenvalue is two at most provided is a bounded convex domain ofR ².Supported in part by NSF DMS 84-09447Home Institution: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA     

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     

Almost periodic Schrödinger operators

We studyH=–d ²/dx ²+V(x) withV(x) limit periodic, e.g.V(x)=a _n cos(x/2 ⁿ ) with a _n <. We prove that for a genericV (and for generica _n in the explicit example), (H) is a Cantor ( nowhere dense, perfect) set. For a dense set, the spectrum is both Cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous.Research partially supported by NSF Grant MCS78-01885On leave from Department of Physics, Princeton UniversityOn leave from Departments of Mathematics and Physics, Princeton University; during 1980–81 Sherman Fairchild Visiting Scholar at Caltech     

String structures and the index of the Dirac-Ramond operator on orbifolds

We discuss the relationship between string structures and the topological class H ⁴(M,Z) on non-simply connected manifolds. We also investigate to what extent the index formula for the Dirac-Ramond operator detects the class, .Supported by the NSF Grant #84-07109 and by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069On leave from the University of Wroclaw, Wroclaw, Poland     

Perturbation about the mean field critical point

We consider two models that are small perturbations of Gaussian or mean field models: the first one is a double well /4⁴ — /2² perturbation of a massless Gaussian lattice field in the weak coupling limit (0, proportional to ). The other consists of a spin 1/2 Ising model with long-range Kac type interactions; the inverse range of the interaction, , is the small parameter. The second model is related to the first one via a sine-Gordon transformation. The lattice ^d has dimensiond3.In both cases we derive an asymptotic estimate to first order (in or ²) on the location of the critical point. Moreover, we prove bounds on the remainder of an expansion in or around the Gaussian or mean field critical points.The appendix, due to E. Speer, contains an extension of Weinberg's theorem on the divergence of Feynman graphs which is used in the proofs.Supported by NSF Grant # MCS 78-01885Supported by NSF Grant # PHY 78-15920     

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.     

The evaluation map in field theory,sigma-models and strings I

The rôle of the evaluation map in anomaly calculations for field theory, sigma-models and strings is investigated. In this paper, anomalies in field theory (with and without a backgrounds connection), are obtained as pull-backs of suitable forms via evaluation maps. The cohomology of the group of gauge transformations is computed in terms of the cohomology of the base manifold and of the cohomology of the structure group. This allows us to clarify the different topological significance of gauge and gravitational anomalies. The relation between locality and universality is discussed and local cohomology is linked to the cohomology of classifying spaces. The problem of combining the locality requirement and the index theorem approach to anomalies is also examined. Anomaly cancellation in field theories derived from superstrings is analyzed and the relevant geometrical constraints are discussed.On leave of absence from Dipartimento di Fisica dell'Universitá di Padova and Istituto Nazionale di Fisica Nucleare, Sezione di PadovaWork supported in part by: Ministero Pubblica Istruzione (research project on Geometry and Physics)On leave of absence from Department of Mathematics, University of North Carolina, Chapel Hill, N.C. 27514. Work supported in part by N.S.F.     

Microwave relaxation and phenomenological damping in thin films

The relation between relaxation timeT, frequency swept resonance linewidth , and phenomenological damping is given by =2/T=(_x+_y), where _x,y = (H ₀+(N _x,y –N _z ) 4M _s ).N _x,y,z are sample demagnetizing factors,H ₀ is the effectivez-directed static field, 4M _s is the saturation induction, and is the gyromagnetic ratio. This fairly simple but general relation shows that the numerical relation between damping and relaxation at a given frequency can be quite different for in-plane and normally magnetized thin films. For thesame loss processes, so thatT andT are equal, is larger than . For permalloy films at 1 GHz, =15 . In addition, the conventional field swept linewidth, H=/, is simply related to only forN _x =N _y . Both and H are geometry dependent and do not provide an intrinsic measure of the relaxation. These results are confirmed by both resonance and transient response experiments. The large values of for large angle switching may also be partially explained by this analysis because the relevant magnetization motion is due to a demagnetizing field normal to the film plane.Visiting scientist on leave fromRaytheon Company, U.S.A. Supported by the Japan Society for the Promotion of Science.     

A remark on null infinity

It is shown that to each choice of conformal factor for the conformal metric on future null Infinity (₊) there is associated an equivalence class of slicings of ₊. Each slicing in turn gives rise to a family of null hypersurfaces in the physical space-time.Supported by NSF Grant No. GP-35773X1.     

Bernoulli property for a one-dimensional system with localized interaction

We consider a one-dimensional system of particles on the half line =[0, ] interacting through elastic collisions among themselves and with a wall at the origin. On the first particle a constant forceE is acting, no external forces act on the other particles. All particles are identical except the first one which has a larger mass. We prove that ifE is such that the Gibbs equilibrium state exists, the corresponding equilibrium dynamical system is a Bernoulli flow.On leave from Dipartimento di Matematica e Fisica, Università di Camerino, Camerino, Italy. Partially supported by NSF Grant DMR-81-14726     

The chiral determinant and the eta invariant

For { _y },y, a one parameter family of invertible Weyl operators of possibly non-zero index acting on spinors over an even dimensional compact manifoldX, we express the phase of the chiral determinant det _– in terms of the invariant of a Dirac operator acting on spinors over ×X.Supported in part by NSF Grant No. PHY-82-15249Supported in part by NSF Grant PHY 8605978 and the Robert A. Welch Foundation     

Normal-dominated singularities in static space-times

We define normal-dominated singularities of static solutions of the Einstein equations and show that a uniquely and invariantly defined structure can be assigned to these singularities. We find for the general solution that the dominant term of the Riemann tensor near the singularity is of Petrov Type N. Except for one special class of solutions, it seems that in general the shear of the null geodesics blows up at the same rate as their convergence near the singularity, in contradistinction to the elementary singularity of Newman and Posadas. We compute the structure for a variety of known static solutions as well as the stationary Kerr-Newman metrics.Supported in part by NSF Grant GP 34639 X.     

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     

MultipletSO(n+1) scalar fields driving eternal expansion in closed (D+1)-dimensional FLRW cosmologies

The possibility of having a de Sitter asymptotic stage free of choice of the value of the positive cosmological constant (no critical ) is analyzed in a closed FLRW universe which starts from a quiescent phase of evolution and ends into a textured phase by taking into account multipletSO(n+1) scalar fields.On leave of absence from Universidade Federal do Rio de Janeiro, RJ, Brazil     

Projections of Gibbs measures may be non-Gibbsian

Consider the + phase of the two dimensional nearest neighbor ferromagnetic Ising model at a temperature belowT _c . Let ₊ be the restriction of this measure to a coordinate axis. We prove that there is no one dimensional translation invariant summable interaction for which ₊ is a Gibbs measure. This is proven by showing that if such an interaction existed, ₊ would have large deviation properties different from those it actually has. Percolation methods are used in the proof.Work supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell and by a NSF grant to Cornell. This work was finished while the author was visiting Rutgers University, being supported by the NSF grant 86-12369