Geometric ergodicity for some space-time max-stable Markov chains

Max-stable processes are central models for spatial extremes. In this paper, we focus on some space-time max-stable models introduced in Embrechts et al. (2016). The processes considered induce discrete-time Markov chains taking values in the space of continuous functions from the unit sphere of $\mathbb{R}^3$ to $(0, \infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the state space is not locally compact, making the classical methodology inapplicable. Instead, we use the fact that the state space is Polish and apply results presented in Hairer (2010)

Resummation for Nonequilibrium Perturbation Theory and Application to Open Quantum Lattices

Lattice models of fermions, bosons, and spins have long served to elucidate the essential physics of quantum phase transitions in a variety of systems. Generalizing such models to incorporate driving and dissipation has opened new vistas to investigate nonequilibrium phenomena and dissipative phase transitions in interacting many-body systems. We present a framework for the treatment of such open quantum lattices based on a resummation scheme for the Lindblad perturbation series. Employing a convenient diagrammatic representation, we utilize this method to obtain relevant observables for the open Jaynes-Cummings lattice, a model of special interest for open-system quantum simulation. We demonstrate that the resummation framework allows us to reliably predict observables for both finite and infinite Jaynes-Cummings lattices with different lattice geometries. The resummation of the Lindblad perturbation series can thus serve as a valuable tool in validating open quantum simulators, such as circuit-QED lattices, currently being investigated experimentally.Comment: 15 pages, 9 figure

Dilepton from Disoriented Chiral Condensates

Disoriented chiral condensates are manifested as long wavelength pionic oscillations and their interaction with the thermal environment can be a significant source of dileptons. We calculate the yield of such dilepton production within the linear sigma model and illustrate the basic features of the dilepton spectrum in a schematic model. We find that the dilepton yield with invariant mass near and below $2m_{\pi}$ due to the soft pion modes can be up to two orders of magnitude larger than the corresponding equilibrium yield. We conclude with a discussion on how this enhancement can be detected by present dilepton experiments.Comment: 15 pages, 9 figs, uses epsf and sprocl style files Contribution to Proceedings, International Workshop on Astro Hadron Physics `Hadrons in Dense Matter', APCTP, Seoul, Korea, October 199

High-Accuracy Calculations of the Critical Exponents of Dyson's Hierarchical Model

We calculate the critical exponent gamma of Dyson's hierarchical model by direct fits of the zero momentum two-point function, calculated with an Ising and a Landau-Ginzburg measure, and by linearization about the Koch-Wittwer fixed point. We find gamma= 1.299140730159 plus or minus 10^(-12). We extract three types of subleading corrections (in other words, a parametrization of the way the two-point function depends on the cutoff) from the fits and check the value of the first subleading exponent from the linearized procedure. We suggest that all the non-universal quantities entering the subleading corrections can be calculated systematically from the non-linear contributions about the fixed point and that this procedure would provide an alternative way to introduce the bare parameters in a field theory model.Comment: 15 pages, 9 figures, uses revte

A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory

The goal of this article is to provide a practical method to calculate, in a scalar theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric phase to perform high-precision calculations of the unsubtracted Green's functions at zero momentum in dimension 3, 4, and 5. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting beta reach a critical value beta_c (with up to 16 significant digits in our actual calculations). We show that the round-off errors on the magnetic susceptibility grow like (beta_c -beta)^{-1} near criticality. We show that the systematic errors (finite truncations and volume) can be controlled with an exponential precision and reduced to a level lower than the numerical errors. We justify the use of the truncation for calculations of the high-temperature expansion. We calculate the dimensionless renormalized coupling constant corresponding to the 4-point function and show that when beta -> beta_c, this quantity tends to a fixed value which can be determined accurately when D=3 (hyperscaling holds), and goes to zero like (Ln(beta_c -beta))^{-1} when D=4.Comment: Uses revtex with psfig, 31 pages including 15 figure

A Two-Parameter Recursion Formula For Scalar Field Theory

We present a two-parameter family of recursion formulas for scalar field theory. The first parameter is the dimension $(D)$. The second parameter ($\zeta$) allows one to continuously extrapolate between Wilson's approximate recursion formula and the recursion formula of Dyson's hierarchical model. We show numerically that at fixed $D$, the critical exponent $\gamma$ depends continuously on $\zeta$. We suggest the use of the $\zeta -$independence as a guide to construct improved recursion formulas.Comment: 7 pages, uses Revtex, one Postcript figur

Understanding molecular representations in machine learning: The role of uniqueness and target similarity

The predictive accuracy of Machine Learning (ML) models of molecular properties depends on the choice of the molecular representation. Based on the postulates of quantum mechanics, we introduce a hierarchy of representations which meet uniqueness and target similarity criteria. To systematically control target similarity, we rely on interatomic many body expansions, as implemented in universal force-fields, including Bonding, Angular, and higher order terms (BA). Addition of higher order contributions systematically increases similarity to the true potential energy and predictive accuracy of the resulting ML models. We report numerical evidence for the performance of BAML models trained on molecular properties pre-calculated at electron-correlated and density functional theory level of theory for thousands of small organic molecules. Properties studied include enthalpies and free energies of atomization, heatcapacity, zero-point vibrational energies, dipole-moment, polarizability, HOMO/LUMO energies and gap, ionization potential, electron affinity, and electronic excitations. After training, BAML predicts energies or electronic properties of out-of-sample molecules with unprecedented accuracy and speed