The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

Asymptotic behavior of densities for two-particle annihilating random walks

Consider the system of particles on ^d where particles are of two types—A andB—and execute simple random walks in continuous time. Particles do not interact with their own type, but when anA-particle meets aB-particle, both disappear, i.e., are annihilated. This system serves as a model for the chemical reactionA+B inert. We analyze the limiting behavior of the densities _A (t) and _B (t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densities _A (0)= _B (0) there is a change in behavior fromd4, where _A (t)= _B (t)C/t ^d /4, tod4, where _A (t)= _B (t)C/tast. For unequal initial densities _A (0)< _B (0), _A (t)e ^–cl ind=1, _A (t)e ^–Ct/logt ind=2, and _A (t)e ^–Ct ind3. The termC depends on the initial densities and changes withd. Techniques are from interacting particle systems. The behavior for this two-particle annihilation process has similarities to those for coalescing random walks (A+AA) and annihilating random walks (A+Ainert). The analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system.     

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference _t – ' _t are reflected in relations between the generators , . Indeed for a von Neumann algebraM with separable predual we show that if _t – '_t 0.28 for smallt, then = 0(+)°^-1 where is an inner automorphism ofM and is a bounded derivation ofM. If the difference _t – ' _t =O(t) ast ; 0, then = + and if _t – ' _t 0.28 for allt then =. We prove analogous results for unitary groups on a Hilbert space andC ₀,C ₀ ^* groups on a Banach space.This paper subsumes an earlier work of the same title which appeared as a report from Z.I.F. der Universität BielefeldWith partial support of the U.S. National Science Foundation     

Proton dipolar spin-lattice relaxation in hexagonal ice

The ultraslow motion of defects in high purity hexagonal H₂O ice has been studied by proton dipolarT _1D measurements in the strong collision limit, using the Jeener technique. The obtained NMR correlation times agree rather well with both the Schottky H₂O diffusion timest _s=r ²/6D and the deuteron correlation times in D₂O ice, suggesting that Schottky rather than interstitial diffusion dominates spin-lattice relaxation in both H₂O and D₂O ice.On leave of absence from University of Ljubljana, Institute J. Stefan.     

Direct observation of the second-order Stark effect of theX-B transition in molecular iodine

The second-order Stark shift of the components of the hyperfine structure of the transition¹ _g ⁺ ( = 0,j = 13, 15) ³ _ou ⁺ ( = 43,j = 12, 16) (of molecular iodine have been studied by means of saturated absorption spectroscopy in an external cell with the I₂ vapour located in an electric field. The anisotropic polarizabilities of the upper and lower levels together with the difference between the isotropic polarizabilities of the levels of the transition have been obtained.     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

Extremal invariant states with a spectrum condition and Borchers' theorem

If the energy spectrum of an extremal invariant state is not the whole real line, it is shown that is either pure or uniquely decomposed into mutually disjoint pure states in the way that =^-1 F _{0 t} dt where is a pure state satisfying = with >0. Next we give a slightly generalized version of Borchers' theorem [1] on the innerness of some automorphism group of a von Neumann algebra with a spectrum condition.     

Dynamic structure factor in a random diffusion model

Let {X _t:0} denote random walk in the random waiting time model, i.e., simple random walk with jump ratew ^–1(X _t), where {w(x):x^d} is an i.i.d. random field. We show that (under some mild conditions) theintermediate scattering function F(q,t)=E ₀ (q^d) is completely monotonic int (E ₀ denotes double expectation w.r.t. walk and field). We also show that thedynamic structure factor S(q, w)=2 ₀ cos(t)F(q, t) exists for 0 and is strictly positive. Ind=1, 2 it diverges as 1/||^1/2, resp. –ln(||), in the limit 0; ind3 its limit value is strictly larger than expected from hydrodynamics. This and further results support the conclusion that the hydrodynamic region is limited to smallq and small such that ||D |q|², whereD is the diffusion constant.     

Differential algebras for quantum groups of type B, C, and D

We study higher order bicovariant differential calculi on the quantum groups O_q(N) and Sp _q (N). We show that the second antisymmetrizer exterior algebra _u is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars biinvariant 1-form. Moreover is central in _u and _u is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra _{s L} is isomorphic to the reflection equation algebra. Let be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.     

The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited

The one-dimensional case of the homogeneous Hamilton–Jacobi and Bernoulli equations S_t S _x ² =0, where S(x, t) is Hamilton's principal function of a free particle and also Bernoulli's momentum potential of a perfect liquid, is considered. Non-elementary solutions are looked for in terms of odd power series in t with x-dependent coefficients and even power series in x with t-dependent coefficients. In both cases, and depending upon initial conditions, unexpected regularities are observed in the terms of these expansions and this suggests that S(^x, ^t) should be written as a product of the elementary solution x²/(2^t) and a function f=f() where =(x, |t|) owing to the symmetry property which is that S(x, –t)=–S(x, t). Requiring that this Ansatz satisfies the said equation and choosing the simplest realization of (x, |t|)=₀ |t/t₀| (x/x ₀)0 with , results in a soluble ordinary differential equation, of first order in u=ln and quadratic in f. This ODE has two fixed points: f=1, obviously, and f=0, a new fixed point which is often called trivial. The phase plane (f_u, f) consists of a family of parabolas, all of which pass through the two fixed points. Explicit solutions of the general case are given close to these fixed points. A one-parameter family of solution is found to emerge from the trivial fixed point with non-trivial initial values S(x, 0). Detailed analyses of these findings will be reported elsewhere, bearing in mind that Bernoulli's equation has to be supplemented by the continuity equation satisfied by the density of the liquid.     

Positive resonances and the limiting amplitude principle

We define the positive resonance points of self-adjoint operators without using the analytical continuation of corresponding resolvents and show that the limiting amplitude principle for the abstract wave equation does not take place in general, if ² = , where is the disturbing frequency and is the resonance point. The asymptotics of corresponding solutions as t are obtained, which imply the growth of the oscillation amplitude as t , 0<<1, or as ln t, t .     

On the asymptotic behaviour of infinite gradient systems

We consider gradient systems of infinitely many particles in one-dimensional space interacting via a positive invariant pair potential with a hard core. The main assumption is that is strictly convex within the rangeR of (whereR is a fixed number ). Under some technical conditions we prove the following theorems: Let the initial distribution be given by a translation invariant point process onR ¹. Then there exists only one extreme equilibrium state with a given intensityI() satisfyingI()R ^–1, and all ergodic initial distributions with an intensityI()R ^–1 converge weakly ast to the extreme equilibrium state with the same intensity.     

Compressible Flow: Turbulence at the Surface

In a study of compressible flow, we have tracked the motion of particles that float on a turbulent body of water. The second moment of longitudinal velocity differences scales as in incompressible flow. However the separation R ²(t) of particle pairs does not vary in time according to the Richardson–Kolmogorov prediction R ²(t)t ³. As expected, the self diffusion d ²(t) shows a crossover between ballistic motion d ²(t)t ² at small t and uncorrelated motion d ²(t)t in the longtime limit.     

Scaling functions of susceptibilities toO(1/n) forn-vector models with axial anisotropy

For an axially anisotropicn-vector model withm = O(n) easy – andn – m = O(n) hard components of the order parameter, we derive the susceptibility r ^–1 along one of the equivalent easy axes and the perpendicular one r ^-1 toO(1/n) of the 1/n-expansion in the disordered phase. The results confirm predictions of the scaling theory, e.g.(g, t)=A t ^– X (B g/t ) and (g, t) =A t ^– X (B g/t ), wheret = T – T _c (g = 0),g is the anisotropy parameter andX, X denote the scaling functions. We evaluate the relevant diagrams toO(1/n) which yield the coefficientsA, A and the critical behaviour of the scaling functions and critical amplitudes explicitly for . The extreme anisotropic case, i.e.m = O(1), is discussed briefly in the large-n limit in comparison with the mean field solution.Parts of this paper were presented at the Frühjahrstagung der Deutschen Physikalischen Gesellschaft in Freudenstadt (May 1974).     

Manifestation of Extracoordination in the Resonance Raman Spectra of Water‐Soluble Cation Co‐Porphyrins

Resonance Raman spectra (RRS) of Co(II) and Co(III)5,10,15,20tetrakis(4Nmethylpyridinium)porphyrin ((Co^II(TmpyP4), and Co^III(TMPyP4)) in aqueous solutions at different pH as well as in organic solvents (methanol, ethanol, DMSO, DMF) are obtained. The increased sensitivity of the oscillation frequencies ₂, ₄, ₈, and ₆ — the markers of the oxidation state of a metal — to the nature of an axial ligand has been revealed. For Co^III(TmpyP4), the shifts of the indicated frequencies in extracoordination have turned out to be twofold larger than those for Co^II(TmpyP4). The spectral effects observed are related to different electron influence of the extraligands on the system of the porphyrin ring. In the case of Co(III)porphyrin, interaction of the d orbitals of the metal and the e _g ^*orbitals of the macrocycle is more efficient since its ionic radius is smaller than for the Co(II)complex. For Co^III(TmpyP4), a linear correlation between the oscillation frequencies ₂, ₄, ₈, and ₆ and the experimental Gutmann parameters characterizing the electronacceptor properties of solvents is found.     

Spatial structure in diffusion-limited two-particle reactions

We analyze the limiting behavior of the densities _A(t) and _B(t), and the random spatial structure(r) = ( _A(t)., _B(t)), for the diffusion-controlled chemical reaction A+Binert. For equal initial densities _B(0) = _b(0) there is a change in behavior fromd 4, where _A(t) = _B(t) C/t^d/4, tod 4, where _A(t) = _b(t) C/t ast ; the termC depends on the initial densities and changes withd. There is a corresponding change in the spatial structure. Ind < 4, the particle types separate with only one type present locally, and , after suitable rescaling, tends to a random Gaussian process. Ind >4, both particle types are, after large times, present locally in concentrations not depending on type or location. Ind=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.     

Convergence of Nelson fiffusions

Let _t, _t ⁿ ,n1, be solutions of Schrödinger equations with potentials form-bounded by –1/2 and initial data inH ¹( ^d ). LetP, P ⁿ ,n1, be the probability measures on the path space =C(₊, ^d ) given by the corresponding Nelson diffusions. We show that if { _t ⁿ } _n1 converges to _t inH ¹( ^d ), uniformly int over compact intervals, then converges to in total variation t0. Moreover, if the potentials are in the Kato classK _d , we show that the above result follows fromH ¹-convergence of initial data, andK _d -convergence of potentials.     

The lattice covering time problem for sites visitedk-times

The problem of the covering time for sites visitedk-times is defined as the mean time taken by a random walk to visit each site of a lattice at leastk times. We performed the investigation using Monte Carlo simulations over one dimensional lattices, ofN sites, with periodic boundary conditions. Two different regions are investigated:Nk1 andkN1. In the former region, we obtain a behaviour of the typet _k/t ₁=a –Bk ^–0.35+A(k)N ^–0.75, (a <1.6). In the latter region we obtain two possible behaviours:t _k k ^0.95 andt _k k(lnk)^–0.5. Two formulas which have a very close behaviour.     

Fock representations of the affine Lie algebraA 1 (1)

The aim of this note is to show that the affine Lie algebraA ₁ ⁽¹⁾ has a natural family _, ,v of Fock representations on the spaceC[x _i,y _j;i andj ], parametrized by (,v) C ². By corresponding the highest weight _, of _, to each (,), the parameter spaceC ₂ forms a double cover of the weight spaceC₀C –₁ with singularities at linear forms of level –2; this number is (–1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA ₁ ⁽¹⁾ -modules for generic (,v).     

Exact distribution of cluster size and perimeter for two-dimensional percolation

Exact series expansion data of Sykes et al. are used to calculate the average numberc _n and perimeters _n of clusters of sizen20 in the site percolation problem for the triangular, square, and honeycomb lattice. At the percolation thresholdp _n we find a sharply peaked distribution of perimeterss _n with mean s _n =((1–p _n )/p _c )n+O(n ) and width s _n ² –S _n ²n ^1.6 where1/=0.39. This perimeter s _n should not be interpreted as a cluster surface in the usual sense. Two tests confirm the universality hypothesis with reasonable accuracy. The asymptotic decay of the cluster numbersc _n withn is consistent with the postulated asymmetry aboutp _c : logc _n –n forn with1 forp<p _c and1/2 forp>p _c .