The Migdal-Kadanoff renormalization group scheme for the Ising model with a free surface

The Migdal-Kadanoff scheme is applied to the Ising model with a free surface. The resulting renormalization group transformation and the duality transformation commute in any dimension. Two simple recursion relations are obtained which reproduce the global phase diagram for the semi-infinite Ising model. The surface critical exponents space methods. In dimensiond=2+, we find the exponentsy t ₁ ^(SB)= andy h ₁ ^(SB)=1+ for the multicritical surface-bulk transition. We also derive and discuss approximate differential recursion relations for the bulk and the surface free energies.     

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].     

Some remarkable degenerations of quantum groups

We show that an irreducible representation of a quantized enveloping algebraU at a ^th root of 1 has maximal dimension (= ^N ) if the corresponding symplectic leaf has maximal dimension (=2N). The method of the proof consists of a construction of a sequence of degenerations ofU , the last one being aq-commutative algebraU ^(2N) . This allows us to reduce many problems concerningU to that concerningU ^(2N) .To Armand Borel on his 70^th birthdaySupported in part by the NSF grant DMS-9103792     

A stochastic theory of adiabatic invariance

LetI be a set of invariants for a system of differential equations with an ordero() vector field. When order perturbations of zero mean are added to the system we show that, under suitable regularity and ergodicity conditions,I becomes an adiabatic invariant with maximal variations of order one on time scales of order 1/². In the stochastically perturbed case,I behaves asymptotically (for small ) like a diffusion process on 1/² time scales. The results also apply to an interesting class of deterministic perturbations. This study extends the results of Khas'minskii on stochastically averaged systems, as well as some of the deterministic methods of averaging, to such invariants.Supported by NSF grant DMR-8704348     

Averaged electron Green function and CDW pinning in one-dimensional random potential

The averaged retarded electron Green functionG ⁺(,k) in 1d disordered metal is calculated using the Berezinsky diagram technique. Using the Gorkov's theory it is shown, that the substitution of inG ⁺ (,k) by the square of the external frequency atk=0 gives the dependence of Fröhlich conductivity _F(). This dependence describes the impurity pinning of CDW in 1d disordered metals. The good agreement of this dependence with experimental data Zeller et al. about _F() in quasi-1d conductor KCP is found     

Particles and scaling for lattice fields and Ising models

The conjectured inequality ⁽⁶⁾0 leads to the existence of _d ⁴ fields and the scaling (continuum) limit ford-dimensional Ising models. Assuming ⁽⁶⁾0 and Lorentz covariance of this construction, we show that ford6 these _d ⁴ fields are free fields unless the field strength renormalizationZ ^–1 diverges. Let be the bare charge and the lattice spacing. Under the same assumptions (⁽⁶⁾0, Lorentz covariance andd6) we show that if ^4–d is bounded as 0, thenZ ^–1 is bounded and the limit field is free.Supported in part by the National Science Foundation under Grant MPS 74-13252Supported in part by the National Science Foundation under Grant MPS 75-21212     

On spherically symmetric perfect fluid distributions and class one property

It is shown that for a spherically symmetric perfect fluid solution to be of class one, either (i) =0, or (ii) +R=0, andR being respectively the eigenvalue of the Weyl tensor in Petrov's classification and spur of the Ricci tensor. Hence, it is deduced that whereas every conformally flat perfect fluid solution is of class one, the converse is not true in general. However, the converse does hold for all solutions with=3p.     

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant (x),xR ³, is considered. The medium is assumed to consist of two components: the background, where (x) = _b , and the embedded component composed of periodically positioned disjoint cubes, where (x) = _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point provided that the size of cubesL, the distancel=L betwen them, and the contrast = _b / _a are chosen in such a way thatL ^–2, and quantities ^-1-3/2 and ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw(^3/2)^-1 of the set {² L ^-2 k ²:kZ³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.     

Dissipation Statistics of a Passive Scalar in a Multidimensional Smooth Flow

We compute analytically the probability distribution function () of the dissipation field =()² of a passive scalar advected by a d-dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor–Kraichnan regime). The tail of the distribution is a stretched exponential: for , ln ()–(d ² )^1/3.     

Random shearing direction models for isotropic turbulent diffusion

Recently, a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long-range spatiotemporal correlations. New random shearing direction models for isotropic turbulent diffusion are introduced here. In these models the velocity field has the spatial second-order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long-range spatial correlations with infrared divergence, but the temporal correlations have finite range. The explicit theory of renormalization for the mean and second-order statistics is developed here. With the spectral parameter, for –<<4 and measuring the strength of the infrared divergence of the spatial spectrum, the scalar mean statistics rigorously exhibit a phase transition from mean-field behavior for <2 to anomalous behavior for with 2<<4 as conjectured earlier by Avellaneda and the author. The universal inertial range renormalization for the second-order scalar statistics exhibits a phase transition from a covariance with a Gaussian functional form for with <2 to an explicit family with a non-Gaussian covariance for with 2<<4. These non-Gaussian distributions have tails that are broader than Gaussian as varies with 2<<4 and behave for large values like exp(–C _c |x|^4–), withC _c an explicit constant. Also, here the attractive general principle is formulated and proved that every steady, stationary, zero-mean, isotropic, incompressible Gaussian random velocity field is well approximated by a suitable superposition of random shear layers.     

Geometric expansion of the boundary free energy of a dilute gas

We consider a dilute classical gas in a volume ^–1 which tends to ^d by dilation as 0. We prove that the pressurep(^–1) isC ^q in at =0 (thermodynamic limit), for anyq, provided the boundary isC ^q and provided the Ursell functionsu _n (x ₁, ...,x _n) admit moments of degreeq and have nice derivatives.     

Borel summability of the unequal double well

Unlike the =0 case, the perturbation series of the unequal double wellp ²+x ²+2gx ³+g ²(1+)x ⁴ are Borel summable to the eigenvalues for any >0.     

The dynamic scattering factor for linear and ring polymers using renomalization group techniques

The dynamic scattering factorS(k,t) for simple ring polymers and linear chains in the presence of hydrodynamic interactions is calculated toO() (4–d, d being the spatial dimensionality) and toO(k ⁴) (k being the external momentum). Results are presented in universal functional form toO().     

The Schrödinger equation and canonical perturbation theory

LetT ₀(, )+V be the Schrödinger operator corresponding to the classical HamiltonianH ₀()+V, whereH ₀() is thed-dimensional harmonic oscillator with non-resonant frequencies =(₁, ... , _d ) and the potentialV(q ₁, ... ,q _d) is an entire function of order (d+1)^–1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT ₀(, ) admits a convergent expansion in powers of of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues _n (, ) ofT ₀+V such thatn coincides with a KAM torus are given, up to order , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in .     

Slow quenching for a one-dimensional kinetic Ising model: Residual energy and domain growth

A one-dimensional kinetic Ising model with Glauber dynamics subjected to a slow continuous quench to zero temperature is studied. For a rather general class of cooling schemes, described by a time-dependent temperatureT(t), the mean domain sizeL(t) is calculated along with the residual energye _res (r) as a function of the cooling rater. If the attempt frequency =₀ exp(–/kT), entering into the transition rates, is temperature dependent (i.e., the barrier is non-zero), the asymptotic growth ofL(t) is given byL()–L(t)~exp[–/kT(t)]. For this case the residual energy exhibits a power-law behaviore _res(r) ~r ^{/2(1 + )} forr small, where =4J/ andJ is the nearest neighbor coupling constant. For =0 and for certain cooling schemes the residual energy is zero andL(t)~t1/2, independent ofr.     

Correlation tensors of the blackbody radiation in rectangular cavities

Electromagnetic equilibrium fluctuations in finite cavities filled with a dissipative medium (dielectric function ()=+i) and bounded by walls of infinite conductivity are considered. Expanding the fields in terms of a complete and orthonormal set of functions and solving the Maxwell equations the response of the EM field to external forces (polarization and magnetization) is obtained. With the aid of the fluctuation dissipation theorem and the linear response functions the 2nd order correlation tensors of the EM field are derived.For rectangular cavities explicit considerations are made. In the case of transparent media (=0) the spectral energy density of the EM radiation is calculated.     

Semi-infinite Potts model and percolation at surfaces

The effects of surfaces on percolation are investigated near the bulk percolation threshold ind=6– dimensions. Using field-theoretic methods, this is done within the framework of a semi-infinite continuousq-state Potts model withq1. Renormalization-group equations are obtained which imply that the usual scaling laws for surface and bulk exponents are valid to all orders in , and the surface exponents at the ordinary and special transition are computed to order . Our result for ₁ ^ord is in conformity with the one by Carton.     

On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping , analytic and -close to the identity, there exists an analytic autonomous Hamiltonian system, H such that its time-one mapping _H differs from by a quantity exponentially small in 1/. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of orders to integrate a Hamiltonian systemK, one actually follows exactly, namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian H, or equivalently of the rescaled Hamiltonian K=^-1H, which differs fromK, but turns out to be ⁵ close to it. Special attention is devoted to numerical integration for scattering problems.     

Construction of analytic KAM surfaces and effective stability bounds

A class of analytic (possibly) time-dependent Hamiltonian systems withd degrees of freedom and the corresponding class of area-preserving, twist diffeomorphisms of the plane are considered. Implementing a recent scheme due to Moser, Salamon and Zehnder, we provide a method that allows us to construct explicitly KAM surfaces and, hence, to give lower bounds on their breakdown thresholds. We, then, apply this method to the HamiltonianHy ²/2+(cosx+cos(x–t)) and to the map (y,x)(y+ sinx,x+y+ sinx) obtaining, with the aid of computer-assisted estimations, explicit approximations (within an error of 10^–5) of the golden-mean KAM surfaces for complex values of with || less or equal than, respectively, 0.015 and 0.65. (The experimental numerical values at which such surfaces are expected to disappear are about, respectively, 0.027 and 0.97.) A possible connection between break-down thresholds and singularities in the complex -plane is pointed out.To our friend and colleague Paola CalderoniSupported by Consiglio Nazionale delle Ricerche, Italy     

Critical behaviour of the compressible magnet with exchange anisotropy

Then-component magnet with exchange anisotropy on a compressible lattice, with isotropic elastic properties, is studied. The renormalization group method is applied ind =4 — dimensions. The fixed points and the stability regions are explored to the order ², and the analysis is concentrated upon the casen<4—2 +O( ²). Investigation of the fixed points reveals various crossover phenomena which are not present in the corresponding rigid model. Renormalization of the anisotropy crossover exponent is demonstrated. It is shown that macroscopic instabilities, leading to the first order phase transition, may appear.