Exact Exponents Research Articles

Exact exponent relations for random resistor-diode networks

Investigates the percolation properties of a random network of non-directed bonds (resistors) and arbitrarily oriented directed bonds (diodes). For the square lattice, there is a multicritical line which connects the isotropic percolation threshold with a network fully occupied by randomly oriented diodes ('random Manhattan'). Along this line, symmetry and invariance properties are used to demonstrate that the correlation length exponents are constant. Furthermore, a lattice independent relation is proved which shows that at the isotropic percolation threshold, the correlation length diverges with the same exponent when the transition is approached by varying either the resistor concentration or the concentration of randomly oriented diodes.

First- and second-order phase transitions in Potts models: Competing mechanisms (invited)

Many condensed matter systems, ranging from adsorbed surfaces to bulk magnets, are microscopically modelled by interacting q-state Potts spins, arrayed in d dimensions. A changeover from second-order phase transitions at q≤qc(d) to first-order transitions at q≳qc can be understood as a condensation of effect vacancies, which are patches of local disorder favored by entropy. Accordingly, the renormalization-group treatment of Potts models is within context of Potts-lattice-gas models, where critical and tricritical fixed points occur at low q, but merge and annihilate at qc. This picture has led to exact tricritical exponents in two dimensions. It is also consistent with recent experimental results on intercalated systems in three dimensions. Effective vacancies in pure Potts models have also been studied by Monte Carlo simulation. Their effective chemical potential can be controlled by a four-point interaction, which proved useful in Monte Carlo renormalization-group studies.

Scaling theory for classical anisotropic Heisenberg chain

The classical anisotropic Heisenberg chain (with Jperpendicular to /J/sub ///= alpha ) is treated by a simple scaling theory valid in the low-temperature limit (K identical to J/sub ////KBT>>1) for any alpha between 0 (Ising) and 1 (isotropic case). The correlation lengths are given in terms of universal single-variable scaling functions Phi /sub //, perpendicular to /: 1/ xi /sub //, perpendicular to /=(cosh-1(1/ alpha )) Phi /sub //, perpendicular to /(K(1- alpha 2)12/+F( alpha )). The term F( alpha ), given in the text, is negligible except when alpha approximately 0. The asymptotic forms of the functions are: Phi /sub ///(x) approximately Phi perpendicular to (x) approximately A/x for x >1 (where A, B and C are constants), giving the characteristic 'Heisenberg-like' and 'Ising-like' behaviours occurring on either side of the crossover which takes place at x approximately 1. Exact critical and crossover exponents are deduced.

Decimation transformations for two-dimensional Ising spin systems

The decimation transformation is used to calculate critical exponents for two-dimensional Ising spin systems on the square, triangular and hexagonal lattices. Both cumulant and cluster approximations are used. While the results of the low-order cumulant and of small cluster calculations agree well with the known exact exponents, an increase in cluster size does not appear to increase accuracy. The decimation transformation is only found to be useful for calculating thermal eigenvalues.

Error exponent for source coding with a fidelity criterion

For discrete memoryless sources with a single-letter fidelity criterion, we study the probability of the event that the distortion exceeds a level <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> , if for large block length the best code of given rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R &gt; R(d)</tex> is used. Lower and upper exponential bounds are obtained, giving the asymptotically exact exponent, except possibly for a countable set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> values.

Observations of the atmospheric Cerenkov light associated with EAS at 5 200 m altitude

An array of four atmospheric Cerenkov light (CL) detectors has been installed on Mt. Chacaltaya (altitude 5 200 m) and operated in conjunction with the Bolivian Air Shower Joint Experiment (BASJE) for a period of several months. The CL detectors are located at radii of 15, 75, 100, and 300 m from the center of the BASJE array. Each detector consists of nine 5-in.-diameter photomultiplier tubes with a total sensitive area of 880 cm2. The angular response for incident photons is 65° FWHM about the vertical. The detector integration time is about 10−7 s, and the associated electronics has a logarithmic response of more than three decades. In the presence of background starlight, the minimum detectable photon pulse is about 104 visible photons/m2. The data exhibit the qualitative features expected for atmospheric CL emission. The average CL intensity varies approximately as N0.8 where the exact exponent depends upon the distance from the shower core. The fluctuations in CL intensity at a fixed distance from the core of EAS of fixed size are found to be greater than the experimental uncertainties. Therefore the observed fluctuations must reflect, in part, the inherent fluctuations in the development of EAS together with the effects of any mixture in the chemical composition of the primary cosmic-ray beam.

Stability of linear periodic systems

A method for calculating the solution, in Floquet form, to a system of linear differential equations with periodic parameters is developed. As a result, both the periodic and exponential parts of the solution are developed as power series in a small parameter, ϵ. From this solution, approximate expressions for the characteristic exponents are derived and hence the stability of the system can be determined. The technique is analogous and parallels that developed by Cesari, Hale, Gambill, et al. By returning to the basis of the underlying process of casting out the secular terms, a method is developed which has advantages from the technical and computational point of view. The expressions for the characteristic exponents are explicit, namely, series of powers of ϵ, whose terms can be easily written, and which converge for ϵ sufficiently small. More importantly, from a practical standpoint, an estimate of the error which results when the series is truncated is derived. Using such an estimate, we draw a circle on the complex plane, centered at the approximate characteristic exponents, within which the exact characteristic exponents must lie.