Decay of Correlations in Ferromagnets

Some new correlation inequalities are described which bound large-distance behavior of correlations in ferromagnets from above by correlations at intermediate distances. Among applications are (1) an inequality, η < 1, on the decay of correlations at the critical point; (2) an inequality χ ≧ coth(1/2 m) relating the zero-field susceptibility and the mass gap in a nearest-neighbor ferromagnet; (3) a finite algorithm for rigorously computing a sequence of upper bounds guaranteed to converge to the true transition temperature in Ising ferromagnets

A proposed study of multiple scattering through clouds up to 1 THz

A rigorous computation of the electromagnetic field scattered from an atmospheric liquid water cloud is proposed. The recent development of a fast recursive algorithm (Chew algorithm) for computing the fields scattered from numerous scatterers now makes a rigorous computation feasible. A method is presented for adapting this algorithm to a general case where there are an extremely large number of scatterers. It is also proposed to extend a new binary PAM channel coding technique (El-Khamy coding) to multiple levels with non-square pulse shapes. The Chew algorithm can be used to compute the transfer function of a cloud channel. Then the transfer function can be used to design an optimum El-Khamy code. In principle, these concepts can be applied directly to the realistic case of a time-varying cloud (adaptive channel coding and adaptive equalization). A brief review is included of some preliminary work on cloud dispersive effects on digital communication signals and on cloud liquid water spectra and correlations

Fast n-point correlation functions and three-point lensing application

We present a new algorithm to rapidly compute the two-point (2PCF), three-point (3PCF) and n-point (n-PCF) correlation functions in roughly O(N log N) time for N particles, instead of O(N^n) as required by brute force approaches. The algorithm enables an estimate of the full 3PCF for as many as 10^6 galaxies. This technique exploits node-to-node correlations of a recursive bisectional binary tree. A balanced tree construction minimizes the depth of the tree and the worst case error at each node. The algorithm presented in this paper can be applied to problems with arbitrary geometry. We describe the detailed implementation to compute the two point function and all eight components of the 3PCF for a two-component field, with attention to shear fields generated by gravitational lensing. We also generalize the algorithm to compute the n-point correlation function for a scalar field in k dimensions where n and k are arbitrary positive integers.Comment: 37 pages, 6 figures, LaTeX; added and modified figures, modified theoretical estimate of computing time; accepted by New Astronom

Efficient Computation of Probabilities of Events Described by Order Statistics and Applications to Queue Inference

This paper derives recursive algorithms for efficiently computing event probabilities related to order statistics and applies the results in a queue inferencing setting. Consider a set of N i.i.d. random variables in [0, 1]. When the experimental values of the random variables are arranged in ascending order from smallest to largest, one has the order statistics of the set of random variables. Both a forward and a backward recursive O(N3 ) algorithm are developed for computing the probability that the order statistics vector lies in a given N-rectangle. The new algorithms have applicability in inferring the statistical behavior of Poisson arrival queues, given only the start and stop times of service of all N customers served in a period of continuous congestion. The queue inference results extend the theory of the "Queue Inference Engine" (QIE), originally developed by Larson in 1990 [8]. The methodology is extended to a third O(N 3 ) algorithm, employing both forward and backward recursion, that computes the conditional probability that a random customer of the N served waited in queue less than r minutes, given the observed customer departure times and assuming first come, first served service. To our knowledge, this result is the first O(N3 ) exact algorithm for computing points on the in-queue waiting time distribution function,conditioned on the start and stop time data. The paper concludes with an extension to the computation of certain correlations of in-queue waiting times. Illustrative computational results are included throughout

Improved Hilbert space exploration algorithms for finite temperature calculations

Computing correlation functions in strongly-interacting quantum systems is one of the most important challenges of modern condensed matter theory, due to their importance in the description of many physical observables. Simultaneously, this challenge is one of the most difficult to address, due to the inapplicability of traditional perturbative methods or the few-body limitations of numerical approaches. For special cases, where the model is integrable, methods based on the Bethe Ansatz have succeeded in computing the spectrum and given us analytical expressions for the matrix elements of physically important operators. However, leveraging these results to compute correlation functions generally requires the numerical evaluation of summations over eigenstates. To perform these summations efficiently, Hilbert space exploration algorithms have been developed which has resulted most notably in the ABACUS library. While this performs quite well for correlations on ground states or low-entropy states, the case of high entropy states (most importantly at finite temperatures or after a quantum quench) is more difficult, and leaves room for improvement. In this work, we develop a new Hilbert space exploration algorithm for the Lieb-Liniger model, specially tailored to optimize the computational order on finite-entropy states for correlations of density-related operators.Comment: 32 pages, 9 figure

Decay of Correlations in Ferromagnets

Some new correlation inequalities are described which bound large-distance behavior of correlations in ferromagnets from above by correlations at intermediate distances. Among applications are (1) an inequality, η < 1, on the decay of correlations at the critical point; (2) an inequality χ ≧ coth(1/2 m) relating the zero-field susceptibility and the mass gap in a nearest-neighbor ferromagnet; (3) a finite algorithm for rigorously computing a sequence of upper bounds guaranteed to converge to the true transition temperature in Ising ferromagnets