Markov chain sampling of the O(n)O(n) loop models on the infinite plane

It was recently proposed in https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.043322 [Herdeiro & Doyon Phys.,Rev.,E (2016)] a numerical method showing a precise sampling of the infinite plane 2d critical Ising model for finite lattice subsections. The present note extends the method to a larger class of models, namely the $O(n)$ loop gas models for $n \in (1,2]$. We argue that even though the Gibbs measure is non local, it is factorizable on finite subsections when sufficient information on the loops touching the boundaries is stored. Our results attempt to show that provided an efficient Markov chain mixing algorithm and an improved discrete lattice dilation procedure the planar limit of the $O(n)$ models can be numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the only requirement for the present numerical method to work.Comment: v2: added conclusion section, changes in introduction and appendice

Monte-Carlo methods for NLTE spectral synthesis of supernovae

We present JEKYLL, a new code for modelling of supernova (SN) spectra and lightcurves based on Monte-Carlo (MC) techniques for the radiative transfer. The code assumes spherical symmetry, homologous expansion and steady state for the matter, but is otherwise capable of solving the time-dependent radiative transfer problem in non-local-thermodynamic-equilibrium (NLTE). The method used was introduced in a series of papers by Lucy, but the full time-dependent NLTE capabilities of it have never been tested. Here, we have extended the method to include non-thermal excitation and ionization as well as charge-transfer and two-photon processes. Based on earlier work, the non-thermal rates are calculated by solving the Spencer-Fano equation. Using a method previously developed for the SUMO code, macroscopic mixing of the material is taken into account in a statistical sense. In addition, a statistical Markov-chain model is used to sample the emission frequency, and we introduce a method to control the sampling of the radiation field. Except for a description of JEKYLL, we provide comparisons with the ARTIS, SUMO and CMFGEN codes, which show good agreement in the calculated spectra as well as the state of the gas. In particular, the comparison with CMFGEN, which is similar in terms of physics but uses a different technique, shows that the Lucy method does indeed converge in the time-dependent NLTE case. Finally, as an example of the time-dependent NLTE capabilities of JEKYLL, we present a model of a Type IIb SN, taken from a set of models presented and discussed in detail in an accompanying paper. Based on this model we investigate the effects of NLTE, in particular those arising from non-thermal excitation and ionization, and find strong effects even on the bolometric lightcurve. This highlights the need for full NLTE calculations when simulating the spectra and lightcurves of SNe.Comment: Accepted for publication by Astronomy & Astrophysic

A Monte Carlo method for critical systems in infinite volume: the planar Ising model

In this paper we propose a Monte Carlo method for generating finite-domain marginals of critical distributions of statistical models in infinite volume. The algorithm corrects the problem of the long-range effects of boundaries associated to generating critical distributions on finite lattices. It uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of "holographic" boundary condition that encodes the presence of an infinite volume beyond it. We check the quality of the distribution obtained in the case of the planar Ising model by comparing various observables with their infinite-plane prediction. We accurately reproduce planar two-, three- and four-point functions of spin and energy operators. We also define a lattice stress-energy tensor, and numerically obtain the associated conformal Ward identities and the Ising central charge.Comment: 43 pages, 21 figure

Zero Variance Markov Chain Monte Carlo for Bayesian Estimators

A general purpose variance reduction technique for Markov chain Monte Carlo (MCMC) estimators, based on the zero-variance principle introduced in the physics literature, is proposed to evaluate the expected value, of a function f with respect to a, possibly unnormalized, probability distribution . In this context, a control variate approach, generally used for Monte Carlo simulation, is exploited by replacing f with a dierent function, ~ f. The function ~ f is constructed so that its expectation, under , equals f , but its variance with respect to is much smaller. Theoretically, an optimal re-normalization f exists which may lead to zero variance; in practice, a suitable approximation for it must be investigated. In this paper, an ecient class of re-normalized ~ f is investigated, based on a polynomial parametrization. We nd that a low-degree polynomial (1st, 2nd or 3rd degree) can lead to dramatically huge variance reduction of the resulting zero-variance MCMC estimator. General formulas for the construction of the control variates in this context are given. These allow for an easy implementation of the method in very general settings regardless of the form of the target/posterior distribution (only dierentiability is required) and of the MCMC algorithm implemented (in particular, no reversibility is needed).Control variates, GARCH models, Logistic regression, Metropolis-Hastings algorithm, Variance reduction

Mixing times of lozenge tiling and card shuffling Markov chains

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste, by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.Comment: 39 pages, 8 figure