ON FINITE DIFFERENCE SCHEMES FOR THE 3-D WAVE EQUATION USING NON-CARTESIAN GRIDS

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Phase Unwrapping and One-Dimensional Sign Problems

Sign problems in path integrals arise when different field configurations contribute with different signs or phases. Phase unwrapping describes a family of signal processing techniques in which phase differences between elements of a time series are integrated to construct non-compact unwrapped phase differences. By combining phase unwrapping with a cumulant expansion, path integrals with sign problems arising from phase fluctuations can be systematically approximated as linear combinations of path integrals without sign problems. This work explores phase unwrapping in zero-plus-one-dimensional complex scalar field theory. Results with improved signal-to-noise ratios for the spectrum of scalar field theory can be obtained from unwrapped phases, but the size of cumulant expansion truncation errors is found to be undesirably sensitive to the parameters of the phase unwrapping algorithm employed. It is argued that this numerical sensitivity arises from discretization artifacts that become large when phases fluctuate close to singularities of a complex logarithm in the definition of the unwrapped phase.Comment: 42 pages, 16 figures. Journal versio

Exact Algorithm for Sampling the 2D Ising Spin Glass

A sampling algorithm is presented that generates spin glass configurations of the 2D Edwards-Anderson Ising spin glass at finite temperature, with probabilities proportional to their Boltzmann weights. Such an algorithm overcomes the slow dynamics of direct simulation and can be used to study long-range correlation functions and coarse-grained dynamics. The algorithm uses a correspondence between spin configurations on a regular lattice and dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson, Proc. 8th Symp. Discrete Algorithms 258, (1997)] for sampling dimer coverings on a planar lattice is adapted to generate samplings for the dimer problem corresponding to both planar and toroidal spin glass samples. This algorithm is recursive: it computes probabilities for spins along a "separator" that divides the sample in half. Given the spins on the separator, sample configurations for the two separated halves are generated by further division and assignment. The algorithm is simplified by using Pfaffian elimination, rather than Gaussian elimination, for sampling dimer configurations. For n spins and given floating point precision, the algorithm has an asymptotic run-time of O(n^{3/2}); it is found that the required precision scales as inverse temperature and grows only slowly with system size. Sample applications and benchmarking results are presented for samples of size up to n=128^2, with fixed and periodic boundary conditions.Comment: 18 pages, 10 figures, 1 table; minor clarification

Unwrapping phase fluctuations in one dimension

Correlation functions in one-dimensional complex scalar field theory provide a toy model for phase fluctuations, sign problems, and signal-to-noise problems in lattice field theory. Phase unwrapping techniques from signal processing are applied to lattice field theory in order to map compact random phases to noncompact random variables that can be numerically sampled without sign or signal-to-noise problems. A cumulant expansion can be used to reconstruct average correlation functions from moments of unwrapped phases, but points where the field magnitude fluctuates close to zero lead to ambiguities in the definition of the unwrapped phase and significant noise at higher orders in the cumulant expansion. Phase unwrapping algorithms that average fluctuations over physical length scales improve, but do not completely resolve, these issues in one dimension. Similar issues are seen in other applications of phase unwrapping, where they are found to be more tractable in higher dimensions.Comment: 14 pages, 7 figures. arXiv admin note: text overlap with arXiv:1806.0183

Sampling of Entire Functions of Several Complex Variables on a Lattice and Multivariate Gabor Frames

We give a general construction of entire functions in $d$ complex variables that vanish on a lattice of the form $L = A (Z + i Z )^d$ for an invertible complex-valued matrix. As an application we exhibit a class of lattices of density >1 that fail to be a sampling set for the Bargmann-Fock space in $C ^2$. By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame

Secure Compute-and-Forward in a Bidirectional Relay

We consider the basic bidirectional relaying problem, in which two users in a wireless network wish to exchange messages through an intermediate relay node. In the compute-and-forward strategy, the relay computes a function of the two messages using the naturally-occurring sum of symbols simultaneously transmitted by user nodes in a Gaussian multiple access (MAC) channel, and the computed function value is forwarded to the user nodes in an ensuing broadcast phase. In this paper, we study the problem under an additional security constraint, which requires that each user's message be kept secure from the relay. We consider two types of security constraints: perfect secrecy, in which the MAC channel output seen by the relay is independent of each user's message; and strong secrecy, which is a form of asymptotic independence. We propose a coding scheme based on nested lattices, the main feature of which is that given a pair of nested lattices that satisfy certain "goodness" properties, we can explicitly specify probability distributions for randomization at the encoders to achieve the desired security criteria. In particular, our coding scheme guarantees perfect or strong secrecy even in the absence of channel noise. The noise in the channel only affects reliability of computation at the relay, and for Gaussian noise, we derive achievable rates for reliable and secure computation. We also present an application of our methods to the multi-hop line network in which a source needs to transmit messages to a destination through a series of intermediate relays.Comment: v1 is a much expanded and updated version of arXiv:1204.6350; v2 is a minor revision to fix some notational issues; v3 is a much expanded and updated version of v2, and contains results on both perfect secrecy and strong secrecy; v3 is a revised manuscript submitted to the IEEE Transactions on Information Theory in April 201

Approximate dual representation for Yang-Mills SU(2) gauge theory

An approximate dual representation for non-Abelian lattice gauge theories in terms of a new set of dynamical variables, the plaquette occupation numbers (PONs) that are natural numbers, is discussed. They are the expansion indices of the local series of the expansion of the Boltzmann factors for every plaquette of the Yang-Mills action. After studying the constraints due to gauge symmetry, the SU(2) gauge theory is solved using Monte Carlo simulations. For a PONs configuration the weight factor is given by Haar-measure integrals over all links whose integrands are products of powers of plaquettes. Herein, updates are limited to changes of the PON at a plaquette or all PONs on a coordinate plane. The Markov chain transition probabilities are computed employing truncated maximal trees and the Metropolis algorithm. The algorithm performance is investigated with different types of updates for the plaquette mean value over a large range of $\beta$s. Using a $12^4$ lattice very good agreement with a conventional heath bath algorithm is found for the strong and weak coupling limits. Deviations from the latter being below 0.1% for $2.5 < \beta < 3$. The mass of the lightest $J^{PC}=0^{++}$ glueball is evaluated and reproduces the results found in the literature