Radius,perimeter, and density profile for percolation clusters and lattice animals

Monte Carlo simulation and series expansion shows the radius of gyration of large clusters withs sites each to vary ass with0.56 in two and0.47 in three dimensions at the percolation threshold, and with(d=2)0.65 and(d=3)0.53 for random lattice animals (zero concentration). Clusters up tos=100 were used. The perimeter of random animals approaches 2.8s for larges on the simple cubic lattice. Monte Carlo simulation of the Eden process (growing animals) up tos=5,000 indicates a systematic variation of about ±0.05 for the effective exponent=(s) and thus suggests that the true asymptotic exponents may be compatible with the predictions of hyper-scaling.     

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.     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Exponential decay in the stark effect

LetH=–+V+Fx ₁ withV(x ₁,x ) analytic in the first variable andV(x ₁+ia, x ) bounded and decreasing to zero asx for eacha . Let be an eigenvector of –+V with negative eigenvalue. Among our results we show that forF0, (,e ^–H ) decays exponentially at a rate governed by the positions of the resonances ofH. This exponential decay is in marked contrast to conventional atomic resonances for which power law decay is the rule.Research supported by NSF Grant No. MCS 78-00101.     

Unbounded derivations and invariant states

Let be a von Neumann algebra with a cyclic and separating vector . Let =i[H, ·] be the spatial derivation implemented by a selfadjoint operatorH, such thatH=0. Let be the modular operator associated with the pair (, ). We prove the equivalence of the following three conditions:1)H is essential selfadjoint onD(), andH commutes strongly with .2) The restriction ofH toD() is essential selfadjoint onD(^1/2) equipped with the inner product(|)_#=(|)+(^1/2|^1/2), , D(^1/2).3) exp (itH) exp (–itH)= for anyt.We show by an example, that the first part of 1),H is essential selfadjoint onD(), does not imply 3). This disproves a conjecture due to Bratteli and Robinson [3].Part of this work was done while O.B. was a member of Zentrum für interdisziplinäre Forschung der Universität Bielefeld     

On the maximization of the central gravitational red-shift of a schwarzschild sphere with a given surface gravitational red-shift

Following Bondi static, spherically symmetric equilibrium configurations with a core and an envelope have been considered. It has been shown that for any configurations with nonnegative pressure and density and with a surface red-shiftz _s 4.77 arbitrarily large central red-shiftsz _c are possible in the limiting case of arbitrarily large radius. The effects of imposition of further constraints in the form of a real speed of sound not exceeding the speed of light are also examined. It is seen that for a given limiting sound-to-light-speed ratio . (i) There exists a limiting surface red-shiftz _s() 1.71. (ii) A configuration withz _s >z _s() is not possible, (iii) A configuration withz _s=z _s() has a unique and finitez _c=z _c(). (iv) Forz _s<z _s() arbitrarily large central red-shifts can be obtained for configurations with arbitrarily large radii.     

A theory of 1/f noise

Letu() be an absolutely integrable function and define the random process where thet _i are Poisson arrivals and thes _i, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum ofn(t), S _n(), in terms of the probability density ofs, p_s(). If any probability density p_s() having the property p_s() I for small is substituted into this formula, the calculated S_n() is such that S_n() 1 for small . However, this is not a spectrum of a well-defined random process; here, it is termed alimit spectrum. If a probability density having the property p_s() for small , where > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter p_s is suitably close to the former p_s, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f ^1– noise (the latter spectrum being integrable for > 0). Suitable examples are given. Actually, u() may be itself a random process, and the theory is developed on this basis.     

Ergodic properties of infinite-dimensional stochastic systems

The ergodic properties of two stochastic models _I and _II are investigated. Each model is described by a fieldx(t),t > 0, on the lattice =Z ^d,d < . For _I,x(t) evolves according to the equations wherex _s (t) R for eachs eF. Here the {w_s(t): s } are independent, one-dimensional Wiener processes, ₂ is a bounded interaction between adjacent lattice sites, and the potentials ₁ and ₂ satisfy appropriate regularity conditions. It is shown that for each model,x(t) is a Markov process on an infinite-dimensional phase spaceX. The probability measures onX that satisfy the Dobrushin-Lanford-Ruelle (DLR) conditions are stationary for this process and have a mixing property. Moreover, for _I any stationary, time-reversal-invariant probability measure that has certain regularity properties must satisfy the DLR conditions.This paper is based on a portion of the author's Ph.D. thesis.⁽²⁾     

Analytic extrapolation inL ∞-norm: An alternative approach to “QCD Sum Rules”

In view of physical applications (especially to QCD Sum Rules), the following problem, pertaining to analytic extrapolation techniques, is studied. We are considering amplitudes, which are (real) analytic functions in the complex plane cut along=[s ₀, ). A modelF ₀(s) of the amplitude is given through the values ofF ₀(s) on some interval=[s ₂,s ₁] (withs ₁<s ₀) and the values of its discontinuity on. These values are approximate, and are supplemented by prescribed error channels, measured inL -norm (both on and). Investigating the compatibility between these data leads to an extremum problem which is solved up to a point where numerical methods can be implemented.Unité Associée au CNRS n^o040768     

Exact exponent for the number of persistent spins in the zero-temperature dynamics of the one-dimensional Potts model

For the zero-temperature Glauber dynamics of theq-state Potts model, the fractionr(q, t) of spins which never flip up to timet decays like a power lawr(q, t)t ^–(q) when the initial condition is random. By mapping the problem onto an exactly soluble one-species coagulation model (A+AA) or alternatively by transforming the problem into a free-fermion model, we obtain the exact expression of (q) for all values ofq. The exponent (q) is in general irrational, (3)=0.53795082..., (4)=0.63151575..., ..., with the exception ofq=2 andq=, for which (2)=3/8 and ()=1.     

New extended model of the Dirac electron

Using a nonlocal function (y)= K (|y –x|)(x) d(x) as a solution of the Dirac equation, we have constructed a new extended model of Dirac's electron. This new model of the electron permits us to eliminate all known divergences in a natural way. The fundamental role of an elementary length (which can be calculated) in the treatment of divergences is presented in detail. The essential feature of the model is the hypothesis of the existence of an electron which possesses a center of charge and a center of mass that do not coincide. Finally, a calculation of the fine structure constant =e ²/hc based on such a model is presented.     

Cantor spectrum and singular continuity for a hierarchical Hamiltonian

We study the spectrum of the HamiltonianH onl ₂() given by (H)(n)=(n+1)+(n–1)+V(n)(n) with the hierarchical (ultrametric) potentialV(2 ^m (2l+1))=(1–R ^m )/(1–R), corresponding to 1-, 2-, and 3-dimensional Coulomb potentials for 0<R<1,R=1 andR>1, respectively, in a suitably chosen valuation metric. We prove that the spectrum is a Cantor set and gaps open at the eigenvaluese _n (1)<e _n (2)<...<e _n (2 ⁿ –1) of the Dirichlet problemH=E, (0)=(2 ⁿ )=0,n1. In the gap opening ate _n (k) the integrated density of states takes on the valuek/2 ⁿ . The spectrum is purely singular continuous forR1 when the potential is unbounded, and the Lyapunov exponent vanishes in the spectrum. The spectrum is purely continuous forR<1 in (H)[–2, 2] and =0 here, but one cannot exclude the presence of eigenvalues near the border of the spectrum. We also propose an explicit formula for the Green's function.Work supported by the Fonds National Suisse de la Recherche Scientifique, Grant No. 2.042-0.86 (H.K. and R.L.) and 2.483-0.87 (A.S.)On leave from the Dipartimento di Fisica, Università degli Studi di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy     

R andRτ ratios at the five-loop level of perturbative QCD

We study perturbative QCD at the five-loop level. In particular we considerR = _tot(e ⁺ e ^– hadrons)/(e ⁺ e ^{– + –}) andR = ( v+hadrons)/( ev). We use our method to estimate the five-loop coefficients. As a result, we obtain _s (M _z ) = 0.1186(11) and _s (34 GeV) = 0.1396(16), which are accurate at the 1% level. We also findR = 3.8350(18), which is consistent withR and is accurate to 0.05%.     

Optical Pumping of Samarium Atoms in a Bichromatic Laser Field in the Presence of Velocity-Changing Collisions

Dark resonances in the ¹⁵⁴Sm -system 4f ⁶6s ²(⁷ F ₀) 4f ⁶6s6p(⁹ F ₁ ⁰) 4f ⁶ s ²(⁷ F ₁) are observed alongside the velocity selective optical pumping. The shape of the resulting spectra strongly depended on the buffer gas (He, Ar) pressure due to velocity-changing collisions (VCC): the sign of the effect could be reversed from the dark to the bright resonance. The observed spectra are interpreted within the framework of the hard-sphere collision model. The role of VCC in the formation of the dark state in the -system is discussed.     

Study ofA+A→0 with probability of reaction and diffusion in one dimension and in fractal substrata

We present Monte Carlo simulations of annihilation reactionA+A0 in one dimensional lattice and in three different fractal substrata. In the model, the particles diffuse independently and when two of them attempt to occupy the same substratum site, they react with a probabilityp. For different kinds of initial distributions and in the short an intermediate time regimes, the results for 0<p1 show that the density ofA particles approximately behaves as (t)=(t=0)f(t/t ₀), with the scaling functionf(x)1 forx1,f(x)x ^–y forx1. The crossover timet ₀, behaves ast _{0 0eff} ^–1y where theeffective initial density _0eff depends on (t=0) and on the kind of initial distribution. For a given substratum of spreading dimensiond _s, the exponenty(d _s/2<y<1) depends only onp and its value increases asp decreases (y1 whenp0). In the very long time regime it is expected thatp(t)t ^–ds/2 independently ofp.     

Effect of hydrogen doping on the magnetic properties of Gd2CuO4

We report the results of ac-susceptibility and dc-magnetization measurements for H_yGd₂CuO₄ (0y0.54). It is shown thatH doping lowers the weak ferromagnetic component in the material. The distinct hysteresis loops observed atT=77 K for both non- and hydrogenated samples change its shape withy. The magnetic ordering temperatures T _N ^Cu and T _N ^Gd , as determined from the temperature dependencies of ac-susceptibility, remain unchanged with sample's hydrogenation. This result seems to indicate that extra electrons are not doped onto the Cu-O planes of Gd₂CuO₄. The frequency dependencies ofx(, T) andx(, T) for bothy=0 andy=0.15 samples are analysed., The maximums ofx andx found at about 200K are considered in terms of susceptibility dependence on the spin-lattice relaxation time (). The anomalies in ac-susceptibility found recently in Gd₂CuO₄ atT _a=8 K andT _b=9.5 K decrease significantly withy. Results are discussed in the context of available data on 214T-type compounds.     

The interface of the Ising model and the Brownian sheet

We study the limit theorem related to the interface of the three-dimensional Ising model. Dobrushin proved that the interface does not fluctuate and becomes rigid for sufficiently large. We define the random fieldX ^L (t, s), 0t, s1, on the interface, and prove that X^L(t, s) converges to the Brownian sheet as L for sufficiently large, whereL denotes the size of the system. This result does not mean that the interface itself converges to the Brownian sheet.     

Learning withQ-state clock neurons

We study the optimal learning capacity for neural networks withQ-state clock neurons, i.e. the states arecomplex numbers with magnitude 1 and azimuthal anglesn·2/Q, withn=0, 1, ...,Q–1. Performing a phase space analysis, the learning capacity _c for given stability can be expressed by means of a double-integral with a simple geometrical interpretation, which for vanishing reduces to _c (Q) = 4Q/(3Q–4), forQ3. Then we define a training algorithm, which generalizes the well-known AdaTron algorithm fromQ=2 toQ3 and converges very fast to the network with optimal stability, if the numberp of random patterns to be learned is smaller than _c (Q). Finally, in the conclusions, we also give hints on applications for image recognition and in a note added in proof we generalize some results to Potts model networks.     

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.     

A new class of long-tailed pausing time densities for the CTRW

We present some asymptotic results for the family of pausing time densities having the asymptotic (t) property(t) [t ln¹⁺(t/T)]^–1. In particular, we show that for this class of pausing time densities the mean-squared displacement r ²(t) is asymptotically proportional to ln(t/T), and the asymptotic distribution of the displacement has a negative exponential form.