3,939 research outputs found
Temporal breakdown and Borel resummation in the complex Langevin method
We reexamine the Parisi-Klauder conjecture for complex e^{i\theta/2} \phi^4
measures with a Wick rotation angle 0 <= \theta/2 < \pi/2 interpolating between
Euclidean and Lorentzian signature. Our main result is that the asymptotics for
short stochastic times t encapsulates information also about the equilibrium
aspects. The moments evaluated with the complex measure and with the real
measure defined by the stochastic Langevin equation have the same t -> 0
asymptotic expansion which is shown to be Borel summable. The Borel transform
correctly reproduces the time dependent moments of the complex measure for all
t, including their t -> infinity equilibrium values. On the other hand the
results of a direct numerical simulation of the Langevin moments are found to
disagree from the `correct' result for t larger than a finite t_c. The
breakdown time t_c increases powerlike for decreasing strength of the noise's
imaginary part but cannot be excluded to be finite for purely real noise. To
ascertain the discrepancy we also compute the real equilibrium distribution for
complex noise explicitly and verify that its moments differ from those obtained
with the complex measure.Comment: title changed, results on parameter dependence of t_c added,
exposition improved. 39 pages, 7 figure
Evolving a puncture black hole with fixed mesh refinement
We present an algorithm for treating mesh refinement interfaces in numerical
relativity. We detail the behavior of the solution near such interfaces located
in the strong field regions of dynamical black hole spacetimes, with particular
attention to the convergence properties of the simulations. In our applications
of this technique to the evolution of puncture initial data with vanishing
shift, we demonstrate that it is possible to simultaneously maintain second
order convergence near the puncture and extend the outer boundary beyond 100M,
thereby approaching the asymptotically flat region in which boundary condition
problems are less difficult and wave extraction is meaningful.Comment: 18 pages, 12 figures. Minor changes, final PRD versio
Evolution of a family of expanding cubic black-hole lattices in numerical relativity
We present the numerical evolution of a family of conformally-flat, infinite,
expanding cubic black-hole lattices. We solve for the initial data using an
initial-data prescription presented recently, along with a new multigrid solver
developed for this purpose. We then apply the standard tools of numerical
relativity to calculate the time development of this initial dataset and derive
quantities of cosmological relevance, such as the scaling of proper lengths.
Similarly to the case of S3 lattices, we find that the length scaling remains
close to the analytical solution for Friedmann-Lemaitre-Robertson-Walker
cosmologies throughout our simulations, which span a window of about one order
of magnitude in the growth of the scale factor. We highlight, however, a number
of important departures from the Friedmann-Lemaitre-Robertson-Walker class.Comment: 20 pages, 15 figures. A few minor correction
Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
This article reviews recent developments in the theoretical understanding and
the numerical implementation of variational renormalization group methods using
matrix product states and projected entangled pair states.Comment: Review from 200
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