Intersection properties of simple random walks: A renormalization group approach

We study estimates for the intersection probability,g(m), of two simple random walks on lattices of dimensiond=4, 4– as a problem in Euclidean field theory. We rigorously establish a renormalization group flow equation forg(m) and bounds on the -function which show that, ind=4,g(m) tends to zero logarithmically as the killing rate (mass)m tends to zero, and that the fixed point,g*, ind=4– is bounded by const' g*const. Our methods also yield estimates on the intersection probability of three random walks ind=3, 3–. For =0, these results were first obtained by Lawler [1].     

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     

Hausdorff Dimension of Sets of Generic Points for Gibbs Measures

For a translation invariant Gibbs measure on the configuration space X of a lattice finite spin system, we consider the set X of generic points. Using a Breiman type convergence theorem on the set X of generic points of an arbitrary translation invariant probability measure on X, we evaluate the Hausdorff dimension of the set X with respect to any metric out of a wide class of scale metrics on X (including Billingsley metrics generated by Gibbs measures).     

Critical behavior of the site percolation model on the square lattice in aL×M geometry

Relevant aspects of the critical behavior of the site percolation model in aL×M geometry (LM) are studied. It is shown that this geometry favors the growth of percolating clusters in theL-direction with respect to those growing in theM-direction, causing pronounced finite-size effects on the percolation probabilities. Scaling functions have an additional parameter, namelyM, which introduces a dependence of these functions on the aspect ratioL/M. At criticality, the probability of a site belonging to the percolation clusters (P _L,M ) behaves likeP _L,M L ^/v (L/M) with =5/36 andv=4/3, where is a suitable scaling function. Using scaling arguments it is conjectured and then tested by means of Monte Carlo simulations, the following asymptotic behavior (L/M)(L/M), (L,M, =1), for the leading term. Systematic deviations of the Monte Carlo data from the conjectured behavior are due to second order corrections to the leading term which can also be under-stood on the basis of scaling ideas. Finite-size dependent critical probabilities are also functions ofL/M as it follows from scaling arguments which are corroborated by the simulations.Financially supported by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) de la República Argentina     

Quantum Ergodicity of Boundary Values of Eigenfunctions

Suppose that ⁿ is a bounded, piecewise smooth domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle B^*() is ergodic. Our proof is based on the classical observation that the boundary values of an interior eigenfunction , =² is an eigenfunction of an operator F_h on the boundary of with h=^–1. In the case of the Neumann boundary condition, F_h is the boundary integral operator induced by the double layer potential. We show that F_h is a semiclassical Fourier integral operator quantizing the billiard map plus a small remainder; the quantum dynamics defined by F_h can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of Zelditch-Zworski (1996). Novelties include the facts that F_h is not unitary and (consequently) the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions. Ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by S. Ozawa (1993), and was almost simultaneously proved by Gerard-Leichtnam (1993) in the case of convex C^1,1 domains (with continuous tangent planes) and with Dirichlet boundary conditions. Our methods seem to be quite different. Motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form.The first author was partially supported by an Australian Research Council Fellowship.The second author was partially supported by NSF grant #DMS-0071358 and DMS-0302518.     

On the relative entropy

ForA any subset of () (the bounded operators on a Hilbert space) containing the unit, and and restrictions of states on () toA, ent _A (|)—the entropy of relative to given the information inA—is defined and given an axiomatic characterisation. It is compared with ent _A ^A (|)—the relative entropy introduced by Umegaki and generalised by various authors—which is defined only forA an algebra. It is proved that ent and ent ^S agree on pairs of normal states on an injective von Neumann algebra. It is also proved that ent always has all the most important properties known for ent ^S : monotonicity, concavity,w* upper semicontinuity, etc.     

The structure of28Si above 10 MeV excitation energy I: gamma-decay modes and radiative widths of levels

The systematics of-decay modes and radiative widths of highly excited states in²⁸Si has been extended by taking-ray spectra on 60 resonances of the²⁷Al(p,) reaction in the range 1097 keVE _p 4492 keV (12643 keVE _x 15915 keV) and on the²⁴Mg(,) resonances atE =3355, 3431, 4003 keV (E _x =12860, 12925, 13 415 keV). The-decay modes of levels in the sub-resonance region (E _x =10–12.5 MeV) were studied with both the²⁷ A1(p, ) reaction and the²⁷ A1(d, n) reaction at E_d=4,5,6 MeV. Information on radiative widths of levels was obtained by measuring the resonance strengthsS =(2I+1) _p / for 52 resonances of the²⁷Al(p, ) reaction with an emphasis on prospective T=1 states.     

<Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">∞</Emphasis></Subscript>-Structures on an Elliptic Curve

The main result of this paper is the proof of the transversal part of the homological mirror symmetry conjecture for an elliptic curve that states an equivalence of two A-categories: one is built using holomorphic vector bundles on an elliptic curve and another is a subcategory in the Fukaya A-category of a torus. The proof is based on the study of A-structures on the category of line bundles over an elliptic curve satisfying some natural restrictions (in particular, m₁ should be zero, m₂ should coincide with the usual composition). The key observation is that such a structure is uniquely determined up to equivalence by certain triple products.This work is partially supported by NSF grant     

Monte Carlo simulation of many-arm star polymers in two-dimensional good solvents in the bulk and at a surface

A Monte Carlo technique is proposed for the simulation of statistical properties of many-arm star polymers on lattices. In this vectorizing algorithm, the length of each arml is increased by one, step by step, from a starting configuration withl=1 orl=2 which is generated directly. This procedure is carried out for a large sample (e.g., 100,000 configurations). As an application, we have studied self-avoiding stars on the square lattice with arm lengths up tol _max=125 and up tof=20 arms, both in the bulk and in the geometry where the center of the star is adsorbed on a repulsive surface. The total number of configurations, which behaves asNl ^{G–1 fl} , where=2.6386 is the usual effective coordination number for self-avoiding walks on the square lattice, is analyzed, and the resulting exponents G=(f) and _s (f) for the bulk and surface geometries are found to be compatible with predictions of Duplantier and Saleur based on conformai invariance methods. We also obtain distribution functions for the monomer density and the distance of the end of an arm from its center. The results are consistent with a scaling theory developed by us.     

Effect of pressure on the behavior of nontransition metallic superconductors

An explicit expression has been derived for the volume dependence of electron-phonon coupling strength () by using a recently proposed quadratic form of the screened pseudopotential and by considering the variation of Fermi momentum with volume. Variation of Coulomb pseudopotential ( ^*) has also been explicitly considered by varying the Fermi momentum. Dependence of transition temperatureT _c and the logarithm of the effective interaction strength (N ₀ V) on the volume has been studied for eight nontransition metallic superconductors. A linear drop inT _c for small values of (–/ ₀) as well as the calculated values of= InN ₀ V/ In, exhibits better matching with the experimental data, as compared to the earlier theoretical results.     

Nontrivial extensions of a representation of the Poincaré group with mass and helicity zero by its tensorial product

Let U(a, ) be a representation of the Poincaré group with mass and helicity zero, realized in the space of C -functions with compact support on ³, without the origin. Let U ⁽²⁾(a, ) denote the tensorial product of U(a, ) by itself. We explicitly determine the cocycles of extension of U(a, ) by U ⁽²⁾(a, ) and we prove that the nontrivial cohomology is indexed by (u(),),u()D ¹ _]0,3\,.     

Matter and light wave interferometry in gravitational fields

We consider the problem of finding the quantum mechanical phase associated with the propagation of a particle in a given external gravitational field, and conclude that it ism ds. In weak fieldsh this allows us to calculate the gravitationally induced phase on a freely traveling particle as 1/2 h P dx whereP is the ordinary momentum. This formula has the expected Newtonian limit and is then used to calculate effects in matter wave interferometry such as those due to gravity waves and the dragging of the ether frame by rotating bodies. Light wave interferometry is then considered and is shown to be also described by 1/2 h K dx , whereK is the wave vector of the light, and the integral is along the path of the ray. Matter and light wave interferometry are compared in various cases.A preliminary version of this work was presented at the Grenoble Workshop on Neutron Interferometry, June 1978.     

Analytic Specific Attenuation Model for Rain for Use in Prediction Methods

Based on the values of k and given by ITU-R for the power-law relationship of specific rain attenuation _R=kR , the analytic expressions for k and are regressed as a function of frequency and are in high agreement with the values tabulated. The specific rain attenuation calculated by the analytic expressions are also in high agreement with those by the values of k and tabulated. Comparisons with other analytic expressions show that the ones in this paper are more accuracy and simple.     

Some jacobi matrices with decaying potential and dense point spectrum

We discuss doubly infinite matrices of the formM _ij= _i,j+1+ _i,j–1+V _{i ij} as operators on ². We present for each >0, examples of potentialsV _n with |V _n|=O(n ^–1/2+) and whereM has only point spectrum. Our discussion should be viewed as a remark on the recent work of Delyon, Kunz, and Souillard.Research partially supported by USNSF under grant MCS 81-20833     

Symplectic Structures on Moduli Spaces of Parabolic Higgs Bundles and Hilbert Scheme

Parabolic triples of the form (E_*,,) are considered, where (E_*,) is a parabolic Higgs bundle on a given compact Riemann surface X with parabolic structure on a fixed divisor S, and is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle (E_*,) a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by d. On the other hand, there is a map from the moduli space of stable parabolic triples to a Hilbert scheme Hilb(Z), where Z denotes the total space of the line bundle K_{X X}(S), that sends a triple (E_*,,) to the divisor defined by the section on the spectral curve corresponding to the parabolic Higgs bundle (E_*,). Using this map and a meromorphic one–form on Hilb(Z), a natural two–form on the moduli space of stable parabolic triples is constructed. It is shown here that this form coincides with the above mentioned form d.     

Retardation of electromagnetic waves by helices of large diameters

This paper is devoted to the study of the wave retardation by helices of large diameters, when the lengthL of one turn of the helix is larger than the vacuum wavelength, i.e.L/>1. The dispersion equation of the tape helix wound up on a dielectric cylinder was derived and the dependence of the phase velocity of slowed-down waves on the diameter of helix was calculated. The theory was verified experimentally at a frequencyf=2·385 GHz. Helices were wound up on polyfoam (=1·04) and plexiglass (=2·55) cylinders of various diameters. For the helix wound up on plexiglass cylinders it was found that the wave retardation depends on the form of conductor from which the helix is made. The retardation of waves is determined by an effective dielectric constant _ef which is equal either ( _i + _e)/2 for the tape helix or ( _{i e}) for the helix made from a wire of the circular cross-section; here _i is the dielectric constant of inner dielectric rod and _e is that of outer space.     

The heat semigroup and integrability of Lie algebras: Lipschitz spaces and smoothness properties

We define and analyze Lipschitz spaces _,q associated with a representationxgV(x) of the Lie algebrag by closed operatorsV(x) on the Banach space together with a heat semigroupS. If the action ofS satisfies certain minimal smoothness hypotheses with respect to the differential structure of (,g,V) then the Lipschitz spaces support representations ofg for which productsV(x)V(y) are relatively bounded by the Laplacian generatingS. These regularity properties of the _,q can then be exploited to obtain improved smoothness properties ofS on . In particularC ₄-estimates on the action ofS automatically implyC -estimates. Finally we use these results to discuss integrability criteria for (,g,V).Dedicated to Res Jost and Arthur Wightman     

Generalized measures in gauge theory

LetP M be a principalG-bundle. We construct well-defined analogs of Lebesgue measure on the spaceA of connections onP and Haar measure on the groupG of gauge transformations. More precisely, we define algebras of cylinder functions on the spacesA,G, andA/G, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from lattice gauge theory, we characterize generalized measures onA,G, andA/G in terms of graphs embedded inM. We use this characterization to construct generalized measures onA andG whenG is compact. The uniform generalized measure onA is invariant under the group of automorphisms ofP. It projects down to the generalized measure onA/G considered by Ashtekar and Lewandowski in the caseG = SU(n). The generalized Haar measure onG is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure onA against generalized Haar measure gives aG-invariant generalized measure onA.     

A poisson—lie structure on the diffeomorphism group of a circle

Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S ¹) of the vector fields on the circleS ¹ and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr X(S ¹) X(S ¹) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S ¹), and a sort of Poisson-Lie structure on the group of diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to quantum diffeomorphism groups.Research supported by the Erwin Schrödinger International Institute for Mathematical Physics.     

Structural relaxation effects on the low-temperature properties of vitreous silica

Measurements of the low-temperature specific heatC and thermal conductivity of vitreous silica after heat treatment at temperaturesT _a between 900°C and 1,400°C are reported. A decrease ofC and an increase of are observed over the whole temperature range studied (C0.06K<T<6K; 0.5 K<T<20 K). Below 1 K the changes inC and (10%) are attributed to a dependence of the density of tunneling states on the fictive temperature. Measurements of the thermal conductivity show that these changes are reversible, thus strongly supporting the evidence for a connection between the tunneling states and the quasi-equilibrium state which is frozen in when an undercooled liquid drops out of thermal equilibrium. Our results are compared to predictions of the free-volume theory of the glass transition. At higher temperaturesC decreases by roughly the same amount as below 1 K while increases by up to 30%. The dependence ofC and onT _a cannot be explained unambigously in terms of a phonon-fraction crossover in the vibrational density of states. Instead, a recently proposed model of coupled SiO₄ rotations is favored.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday