Group duality and the Kubo-Martin-Schwinger condition

We consider clusteringG-invariant states of aC*-algebraU endowed with an action of a locally compact abelian groupG. Denoting as usual byF _AB,G _AB, the corresponding two-point functions, we give criteria for the fulfillment of the KMS condition (w.r.t. some one-parameter subgroup ofG) based upon the existence of a closable mapT such thatTF _AB =G _AB for allA,BU. Closability is either inL (G),B(G), orC (G), according to clustering assumptions. Our criteria originate from the combination of duality results for the groupG (phrased in terms of functions systems), with density results for the two-point functions.Supported in part by the National Science Foundation     

Divergence of the bulk resistance at criticality in disordered media

Consider the electrical resistancer _n (p) of a hypercubic bond lattice [O,n] ^d inZ ^d , where the bonds have resistance 1 with probabilityp or with probability 1-p. Letp _n (p)=n ^2-d r _n (p) andp(p)=lim_np_n(p). It is well known thatp(p)< ifp>p _c andp(p)= ifp<p _c , wherep _c is the percolation threshold. Here we show thatp(p _c )=, and .     

A new method for constructing factorisable representations for current groups and current algebras

LetC _e (R ⁿ ,G) denote the group of infinitely differentiable maps fromn-dimensional Euclidean space into a simply connected and connected Lie group, which have compact support. This paper introduces a class of factorisable unitary representations ofC _e (R ⁿ ,G) with the property that the unitary operatorU _f corresponding to a functionf inC _e (R ⁿ ,G) depends not only onf, but also on the derivatives off up to a certain order. In particular these representations can not be extended to the group of all continuous functions fromR ⁿ toG with compact support.     

A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices

Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X*X(X ^t X) converges to the Tracy–Widom law as n,p (the dimensions of X) tend to in some ratio n/p>0. We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n–p=O(p ^1/3). We also prove that _max(n ^1/2+p ^1/2)²+O(p ^1/2 log(p)) (a.e.) for general >0.     

Almost Sure Weak Convergence for the Generalized Orthogonal Ensemble

The generalized orthogonal ensemble satisfies isoperimetric inequalities analogous to the Gaussian isoperimetric inequality, and an analogue of Wigner's law. Let v be a continuous and even real function such that V(X)=tracev(X)/n defines a uniformly p-convex function on the real symmetric n×n matrices X for some p2. Then (dX)=e ^–V(X) dX/Z satisfies deviation and transportation inequalities analogous to those satisfied by Gaussian measure^{(6, 27)}, but for the Schatten c ^p norm. The map, that associates to each XM ^s _n () its ordered eigenvalue sequence, induces from a measure which satisfies similar inequalities. It follows from such concentration inequalities that the empirical distribution of eigenvalues converges weakly almost surely to some non-random compactly supported probability distribution as n.     

On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transform was introduced by D. Arnal and J. C. Cortet,:L(G) C (V,L(^2d )), whereL(G) is the Schwartz space ofG andV × ^2k is aG-invariant Zariski open set ing ^* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.     

Conservation of Energy,Entropy Identity,and Local Stability for the Spatially Homogeneous Boltzmann Equation

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f ₀(v)(1+|v|²+|logf ₀(v)|)L ¹(R ³), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class L ([0, ); L ¹ ₂(R ³))C ¹([0, ); L ¹(R ³)) [where L ¹ _s (R ³)={ff(v)(1+|v|²) ^s/2L ¹(R ³)}], and in this class, the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy. Moreover, for hard potentials and the hard-sphere model, a local stability result for conservative solutions (i.e., satisfying the conservation of mass, momentum, and energy) is obtained. As an application of the local stability, a sufficient and necessary condition on the initial data f ₀ such that the conservative solutions f belong to L ¹ _loc([0, ); L ¹ ₂₊ (R ³)) is also given.     

Crossover scaling function for the free energy

A cubic field, coupling tos|s|², inn-component spin models induces a bicritical crossover fromn-isotropic to Ising like (m=1) critical behaviour for 1<n<, but to classical behaviour in the limitn. By following the analysis of Nelson and Domany, the bicritical scaling function for the free energy ind dimensions is obtained correct to order =4–d and for general (m,n). The mechanism responsible for the breakdown of hyperscaling in the classical behaviour is discussed.     

Random walk with persistence

The exact analytic result is obtained for the Fourier transform of the generating functionF(R,s)= _n=0 s ⁿ P(R,n), whereP(R,n) is the probability density for the end-to-end distanceR inn steps of a random walk with persistence. The moments R ²(n), R ⁴(n), and R ⁶(n) are calculated and approximate results forP(R,n) and R ^–1(n) are given.     

On the bundle of connections and the gauge orbit manifold in Yang-Mills theory

In an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations (in Yang-Mills theory) is aC principal fibration. The underlying quotient space, the gauge orbit space, is seen explicitly to be aC manifold modelled on a Hilbert space.     

Inverse Problem for a General Bounded Domain inR3 with Piecewise Smooth Mixed BoundaryConditions

We study the influence of a finite container on an ideal gas. The trace of theheat kernel (t) = _{= 1}exp(–t), where {} _{= 1}are the eigenvalues of the negative Laplacian – ² = – ³ _{= 1}(/x )² in the (x ¹, x ², x ³)-space,is studied for a general bounded domain with a smooth bounding surface S, where afinite number of Dirichlet, Neumann, and Robin boundary conditions on thepiecewise smooth parts S _i (i = 1, ..., n) of S are considered such that S =U _{i = 1} S _i . Some geometrical properties of (the volume, the surface area, the meancurvature, and the Gaussian curvature) are determined. Furthermore,thermodynamic quantities, particularly the energy, for an ideal gas enclosed inthe general bounded domain with Dirichlet, Neumann, and Robin conditionsare examined with the help of the asymptotic expansions of (t) for short timet. We show that these thermodynamic quantities depend on some geometricproperties of .     

Equivalence of deformations and associated *-products

In this letter we study the non-trivial formal differentiable deformations of the Lie algebraN=C (W, IR) whereW is a symplectic manifold. Under some assumptions (satisfied in particular forW=IR²ⁿ ) we show that these deformations are all equivalent, up to a monomial change of the parameter, to one of them (Moyal for IR²ⁿ ). Furthermore, if there exists a differentiable *-product corresponding to one of them, each of them is induced by a *-product which is essentially unique.Aspirant du Fonds National belge de la Recherche Scientifique.     

Group duality and the Kubo-Martin Schwinger condition. II

Let be an invariant state of theC*-system { ,G, } on a locally compact noncommutative groupG. Assume further that is extremal -invariant for an action of an amenable groupH which is -asymptotically abelian and commutes with . Denoting byF _AB,G _AB the corresponding two point functions, we give criteria for the fulfillment of the KMS condition with respect to some one parameter subgroup of the center ofG based on the existence of a closable mapT such thatTF _AB=G _AB for allA,B . Closability is either inL (G),B(G) orC (G), according to clustering properties for . The basic mathematical technique is the duality theory for noncompact, noncommutative locally compact groups.This work is supported in part by the National Science Foundation, Grant MCS 79-03041     

Self-avoiding walks and trees in spread-out lattices

LetG _R be the graph obtained by joining all sites ofZ ^d which are separated by a distance of at mostR. Let (G _R ) denote the connective constant for counting the self-avoiding walks in this graph. Let (G _R ) denote the coprresponding constant for counting the trees embedded inG _R . Then asR, (G _R ) is asymptotic to the coordination numberk _R ofG _R , while (G _R ) is asymptotic toek _R. However, ifd is 1 or 2, then (G _R )-k _R diverges to –.Dedicated to Oliver Penrose on this occasion of his 65th birthday.     

Superderivations ofC *-algebras implemented by symmetric operators

The paper studies unbounded symmetric and dissipative implementations (S,G) of^*-superderivations ofC ^*-algebras . It associates with them representations _S of the domainsD() of on the deficiency spacesN(S) of the symmetric operatorsS. A link is obtained between the deficiency indicesn _±(S) ofS and the dimensions of irreducible representations of . For the case when (S,G) is a maximal implementation and max(n _±(S))<, some conditions are given for the representation _S to be semisimple and to extend to a bounded representation of .     

Hamiltonian Structures on Coadjoint Orbits of Semidirect Product G = Diff+(S1) ? < C∞(S1, ?)

We consider the semidirect product of diffeomorphisms of the circle D=Diff₊(S ¹) and C (S ¹, ) functions, classify its coadjoint orbits and treat dynamical systems, related to corresponding Lie algebra centrally extended by the Kac-Moody, Virasoro and semidirect product cocycles with arbitrary coefficients. The three-Hamiltonian (in the case of the generalized DWW-type models) and bi-Hamiltonian (for KdV-type models) structures are found and used in the proof of their complete integrability.     

Nelson's symmetry and the infinite volume behavior of the vacuum inP(φ)2

LetH _l be the Hamiltonian in aP()₂ theory with sharp space cutoff in the interval (–l/2,l/2). LetE _l =inf(H _l ), (l)=–E _l /l, and let _l be the vacuum forH _l . discuss properties of (l) and _l . In particular, asl, there are finite constants <0 and such that (l), ((l)–)l, and hence (l)=+/l+o(l ^–1). Moreover exp(–c ₁ l) _{l 1}exp(–c ₂ l) forc ₁,c ₂ positive constants, where _{l 1} is theL ¹(Q, d₀) norm of ₁ with respect to the Fock vacuum measure. We also present a new proof of recent estimates of Glimm and Jaffe on local perturbations ofH _l in the infinite volume limit.Research sponsored by AFOSR under Contract No. F44620-71-C-0108.On leave from Istituto di Fisica Teorica, Universitá di Napoli and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli.A. Sloan Foundation Fellow.     

Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Let denote the field of algebraic numbers in A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ M_d((G, σ)), regarded as an operator on l²(G)^d, the eigenvalues of A are algebraic numbers, where σ ∈ Z²(G, ) is an algebraic multiplier, and denotes the unitary elements of . Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ. In the special case when σ is rational (σⁿ=1 for some positive integer n) this property holds for a larger class of groups containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators. The second and third authors acknowledge support from the Australian Research Council.     

Existence of solutions of the Robinson-Trautman equation and spatial infinity

The existence of solutions of the Robinson-Trautman equation is established. If solutions exist global in time, they describe spacetimes with negative ADM mass andC scri^±.     

Moyal Deformations of Gravity via SU(∞) Gauge Theories, Branes and Topological Chern-Simons Matrix Models

Moyal noncommutative star-product deformations of higher-dimensional gravitational Einstein-Hilbert actions via lower-dimensional SU(), W gauge theories are constructed explicitly based on the holographic reduction principle. New reparametrization invariant p-brane actions and their Moyal star product deformations follows. It is conjectured that topological Chern-Simons brane actions associated with higher-dimensional knots have a one-to-one correspondence with topological Chern-Simons Matrix models in the large N limit. The corresponding large N limit of Topological BF Matrix models leads to Kalb-Ramond couplings of antisymmetric-tensor fields to p-branes. The former Chern-Simons branes display higher-spin W symmetries which are very relevant in the study of W Gravity, the Quantum Hall effect and its higher-dimensional generalizations. We conclude by arguing why this interplay between condensed matter models, higher-dimensional extensions of the Quantum Hall effect, Chern-Simons Matrix models and Gravity needs to be investigated further within the framework of W Gauge theories.