105,996 research outputs found
Observations Outside the Light-Cone: Algorithms for Non-Equilibrium and Thermal States
We apply algorithms based on Lieb-Robinson bounds to simulate time-dependent
and thermal quantities in quantum systems. For time-dependent systems, we
modify a previous mapping to quantum circuits to significantly reduce the
computer resources required. This modification is based on a principle of
"observing" the system outside the light-cone. We apply this method to study
spin relaxation in systems started out of equilibrium with initial conditions
that give rise to very rapid entanglement growth. We also show that it is
possible to approximate time evolution under a local Hamiltonian by a quantum
circuit whose light-cone naturally matches the Lieb-Robinson velocity.
Asymptotically, these modified methods allow a doubling of the system size that
one can obtain compared to direct simulation. We then consider a different
problem of thermal properties of disordered spin chains and use quantum belief
propagation to average over different configurations. We test this algorithm on
one dimensional systems with mixed ferromagnetic and anti-ferromagnetic bonds,
where we can compare to quantum Monte Carlo, and then we apply it to the study
of disordered, frustrated spin systems.Comment: 19 pages, 12 figure
Asymptotology of Chemical Reaction Networks
The concept of the limiting step is extended to the asymptotology of
multiscale reaction networks. Complete theory for linear networks with well
separated reaction rate constants is developed. We present algorithms for
explicit approximations of eigenvalues and eigenvectors of kinetic matrix.
Accuracy of estimates is proven. Performance of the algorithms is demonstrated
on simple examples. Application of algorithms to nonlinear systems is
discussed.Comment: 23 pages, 8 figures, 84 refs, Corrected Journal Versio
Semi-classical analysis of the inner product of Bethe states
We study the inner product of two Bethe states, one of which is taken
on-shell, in an inhomogeneous XXX chain in the Sutherland limit, where the
number of magnons is comparable with the length L of the chain and the magnon
rapidities arrange in a small number of macroscopically large Bethe strings.
The leading order in the large L limit is known to be expressed through a
contour integral of a dilogarithm. Here we derive the subleading term. Our
analysis is based on a new contour-integral representation of the inner product
in terms of a Fredholm determinant. We give two derivations of the sub-leading
term. Besides a direct derivation by solving a Riemann-Hilbert problem, we give
a less rigorous, but more intuitive derivation by field-theoretical methods.
For that we represent the Fredholm determinant as an expectation value in a
Fock space of chiral fermions and then bosonize. We construct a collective
field for the bosonized theory, the short wave-length part of which may be
evaluated exactly, while the long wave-length part is amenable to a
expansion. Our treatment thus results in a systematic 1/L expansion of
structure factors within the Sutherland limit.Comment: 22 pages, 0 figure
Fast MCMC sampling for Markov jump processes and extensions
Markov jump processes (or continuous-time Markov chains) are a simple and
important class of continuous-time dynamical systems. In this paper, we tackle
the problem of simulating from the posterior distribution over paths in these
models, given partial and noisy observations. Our approach is an auxiliary
variable Gibbs sampler, and is based on the idea of uniformization. This sets
up a Markov chain over paths by alternately sampling a finite set of virtual
jump times given the current path and then sampling a new path given the set of
extant and virtual jump times using a standard hidden Markov model forward
filtering-backward sampling algorithm. Our method is exact and does not involve
approximations like time-discretization. We demonstrate how our sampler extends
naturally to MJP-based models like Markov-modulated Poisson processes and
continuous-time Bayesian networks and show significant computational benefits
over state-of-the-art MCMC samplers for these models.Comment: Accepted at the Journal of Machine Learning Research (JMLR
Kronecker Graphs: An Approach to Modeling Networks
How can we model networks with a mathematically tractable model that allows
for rigorous analysis of network properties? Networks exhibit a long list of
surprising properties: heavy tails for the degree distribution; small
diameters; and densification and shrinking diameters over time. Most present
network models either fail to match several of the above properties, are
complicated to analyze mathematically, or both. In this paper we propose a
generative model for networks that is both mathematically tractable and can
generate networks that have the above mentioned properties. Our main idea is to
use the Kronecker product to generate graphs that we refer to as "Kronecker
graphs".
First, we prove that Kronecker graphs naturally obey common network
properties. We also provide empirical evidence showing that Kronecker graphs
can effectively model the structure of real networks.
We then present KronFit, a fast and scalable algorithm for fitting the
Kronecker graph generation model to large real networks. A naive approach to
fitting would take super- exponential time. In contrast, KronFit takes linear
time, by exploiting the structure of Kronecker matrix multiplication and by
using statistical simulation techniques.
Experiments on large real and synthetic networks show that KronFit finds
accurate parameters that indeed very well mimic the properties of target
networks. Once fitted, the model parameters can be used to gain insights about
the network structure, and the resulting synthetic graphs can be used for null-
models, anonymization, extrapolations, and graph summarization
Pade approximants of random Stieltjes series
We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 +
>...))) where the s_n are independent random variables with the same gamma
distribution. For every realisation of the sequence, S(t) defines a Stieltjes
function. We study the convergence of the finite truncations of the continued
fraction or, equivalently, of the diagonal Pade approximants of the function
S(t). By using the Dyson--Schmidt method for an equivalent one-dimensional
disordered system, and the results of Marklof et al. (2005), we obtain explicit
formulae (in terms of modified Bessel functions) for the almost-sure rate of
convergence of these approximants, and for the almost-sure distribution of
their poles.Comment: To appear in Proc Roy So
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