61 research outputs found
Strong-coupling perturbation theory for the two-dimensional Bose-Hubbard model in a magnetic field
The Bose-Hubbard model in an external magnetic field is investigated with
strong-coupling perturbation theory. The lowest-order secular equation leads to
the problem of a charged particle moving on a lattice in the presence of a
magnetic field, which was first treated by Hofstadter. We present phase
diagrams for the two-dimensional square and triangular lattices, showing a
change in shape of the phase lobes away from the well-known power-law behavior
in zero magnetic field. Some qualitative agreement with experimental work on
Josephson-junction arrays is found for the insulating phase behavior at small
fields.Comment: 7 pages, 5 figures include
Modified perturbation theory approach for tt-bar production and decay
The modified perturbation theory (MPT), based on direct expansion of
probabilities instead of amplitudes, allows one to avoid divergences in the
phase-space integrals resulting from production and decay of unstable
particles. In the present paper the range of applicability of MPT is determined
numerically in the case of the process . It is shown that with the complete expansion in powers of
the coupling constant (without Dyson resummation) MPT operates best at the
energies located near the maximum of the cross-section and slightly above the
maximum. In this region the MPT expansion within the next-to-leading order
considerably exceeds in accuracy well-known DPA approach.Comment: LaTeX, 11 pages, 3 eps figure
Distribution of Eigenvalues for the Modular Group
The two-point correlation function of energy levels for free motion on the
modular domain, both with periodic and Dirichlet boundary conditions, are
explicitly computed using a generalization of the Hardy-Littlewood method. It
is shown that ion the limit of small separations they show an uncorrelated
behaviour and agree with the Poisson distribution but they have prominent
number-theoretical oscillations at larger scale. The results agree well with
numerical simulations.Comment: 72 pages, Latex, the fiogures mentioned in the text are not vital,
but can be obtained upon request from the first Autho
Using Hierarchical Centering to Facilitate a Reversible Jump MCMC Algorithm for Random Effects Models
The first author was supported by a studentship jointly funded by the University of St Andrews and EPSRC, through the National Centre for Statistical Ecology (EPSRC grant EP/C522702/1), with subsequent funding from EPSRC/NERC grant EP/I000917/1.Hierarchical centering has been described as a reparameterization method applicable to random effects models. It has been shown to improve mixing of models in the context of Markov chain Monte Carlo (MCMC) methods. A hierarchical centering approach is proposed for reversible jump MCMC (RJMCMC) chains which builds upon the hierarchical centering methods for MCMC chains and uses them to reparameterize models in an RJMCMC algorithm. Although these methods may be applicable to models with other error distributions, the case is described for a log-linear Poisson model where the expected value λλ includes fixed effect covariates and a random effect for which normality is assumed with a zero-mean and unknown standard deviation. For the proposed RJMCMC algorithm including hierarchical centering, the models are reparameterized by modelling the mean of the random effect coefficients as a function of the intercept of the λλ model and one or more of the available fixed effect covariates depending on the model. The method is appropriate when fixed-effect covariates are constant within random effect groups. This has an effect on the dynamics of the RJMCMC algorithm and improves model mixing. The methods are applied to a case study of point transects of indigo buntings where, without hierarchical centering, the RJMCMC algorithm had poor mixing and the estimated posterior distribution depended on the starting model. With hierarchical centering on the other hand, the chain moved freely over model and parameter space. These results are confirmed with a simulation study. Hence, the proposed methods should be considered as a regular strategy for implementing models with random effects in RJMCMC algorithms; they facilitate convergence of these algorithms and help avoid false inference on model parameters.PostprintPeer reviewe
A review of spatial causal inference methods for environmental and epidemiological applications
The scientific rigor and computational methods of causal inference have had
great impacts on many disciplines, but have only recently begun to take hold in
spatial applications. Spatial casual inference poses analytic challenges due to
complex correlation structures and interference between the treatment at one
location and the outcomes at others. In this paper, we review the current
literature on spatial causal inference and identify areas of future work. We
first discuss methods that exploit spatial structure to account for unmeasured
confounding variables. We then discuss causal analysis in the presence of
spatial interference including several common assumptions used to reduce the
complexity of the interference patterns under consideration. These methods are
extended to the spatiotemporal case where we compare and contrast the potential
outcomes framework with Granger causality, and to geostatistical analyses
involving spatial random fields of treatments and responses. The methods are
introduced in the context of observational environmental and epidemiological
studies, and are compared using both a simulation study and analysis of the
effect of ambient air pollution on COVID-19 mortality rate. Code to implement
many of the methods using the popular Bayesian software OpenBUGS is provided
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