Self-avoiding walk in five or more dimensions I. The critical behaviour

We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford5. We prove that the numberc _n ofn-step self-avoiding walks satisfiesc _n ~A ⁿ , where is the connective constant (i.e. =1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forc _n(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (^––Z)^1/2. The critical two-point function is shown to decay at least as fast as x^–2, and its Fourier transform is shown to be asymptotic to a multiple ofk ^–2 ask0 (i.e. =0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.Supported by the Nishina Memorial Foundation and NSF grant PHY-8896163.Supported by NSERC grant A9351     

Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ ^d × ℤ₊, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ ⁿ (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ ^d × ℤ₊, summing this probability over x  ℤ ^d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ_∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n ⁻¹, we prove existence of a limiting measure ℚ_∞, with ℚ_∞ = ℙ_∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ_∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1. Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002 RID="*" ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl     

Conductivity variations in composites of α-zirconium phosphate and fumed silica

Composite proton-conducting solid electrolytes have been formed from α-zirconium phosphate (α-Zr(HPO₄)₂·H₂O, α-ZrP) and high surface area non-porous fumed silica in varying mole ratios. Conductivity variations as a function of temperature and relative humidity have been characterised and compared to that for a similarly prepared delaminated ZrP (no silica). Conductivities increase with relative humidity and E_a decreases at high silica mole fractions. Conduction is via exposed lamellar surfaces and other microstructural imperfections in the composites.     

Structure determination and absolute configuration of cannabichromanone derivatives from high potency Cannabis sativa

Three new cannabichromanone derivatives were isolated from high potency cannabis, along with the known cannabichromanone. Full spectroscopic data, including the use of electronic circular dichroism and Mosher ester analysis to determine the absolute configuration of these compounds, are reported. All isolates were tested for antimicrobial, antimalarial, antileishmanial, and anti-oxidant activities.     

Li+ transport in Li0.40MoO3 studied by pulse NMR

Pulsed ⁷Li NMR relaxation time measurements are reported for the intercalate Li_0.40MoO₃ in the temperature range 200K < T < 460K at 20 MHz. Temperature dependences of T₁ and T₂ indicate translation of Li⁺ with E_A = 23 kJ mol^-1. The data lead to an estimate for the Li⁺ self-diffusion coefficient at 300K of D¹_300K ? 4 x 10^-12 cm² sec^-1.     

Anomalous optical response with variation in band filling in linear chain conductors

We report the polarized infrared reflectance of ($\text{N}$-methylphenazinium)_x (phenazine)_1-x(tetracyanoquinodimethanide), (NMP)_x(Phen)_1-x(TCNQ) for 0.5≤x≤1.0. We demonstrate that the “Drude edge” in TCNQ salts is related to the conduction electron charge density. An anomalous rapid red shift of the plasma frequency, observed for $\text{x ～}\text{2}\text{3}$, may signal the emptying of the conduction electrons from half of the chains and/or a crossover from small U/W (coulomb repulsion/bandwidth) to large U/W behavior.