8,671 research outputs found
Markov cubature rules for polynomial processes
We study discretizations of polynomial processes using finite state Markov
processes satisfying suitable moment matching conditions. The states of these
Markov processes together with their transition probabilities can be
interpreted as Markov cubature rules. The polynomial property allows us to
study such rules using algebraic techniques. Markov cubature rules aid the
tractability of path-dependent tasks such as American option pricing in models
where the underlying factors are polynomial processes.Comment: 29 pages, 6 Figures, 2 Tables; forthcoming in Stochastic Processes
and their Application
Matrix permanent and quantum entanglement of permutation invariant states
We point out that a geometric measure of quantum entanglement is related to
the matrix permanent when restricted to permutation invariant states. This
connection allows us to interpret the permanent as an angle between vectors. By
employing a recently introduced permanent inequality by Carlen, Loss and Lieb,
we can prove explicit formulas of the geometric measure for permutation
invariant basis states in a simple way.Comment: 10 page
Equilibrium equation of state of a hard sphere binary mixture at very large densities using replica exchange Monte-Carlo simulations
We use replica exchange Monte-Carlo simulations to measure the equilibrium
equation of state of the disordered fluid state for a binary hard sphere
mixture up to very large densities where standard Monte-Carlo simulations do
not easily reach thermal equilibrium. For the moderate system sizes we use (up
to N=100), we find no sign of a pressure discontinuity near the location of
dynamic glass singularities extrapolated using either algebraic or simple
exponential divergences, suggesting they do not correspond to genuine
thermodynamic glass transitions. Several scenarios are proposed for the fate of
the fluid state in the thermodynamic limit.Comment: 10 pages, 8 fig
Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes
When solving stochastic partial differential equations (SPDEs) driven by
additive spatial white noise, the efficient sampling of white noise
realizations can be challenging. Here, we present a new sampling technique that
can be used to efficiently compute white noise samples in a finite element
method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit
the finite element matrix assembly procedure and factorize each local mass
matrix independently, hence avoiding the factorization of a large matrix.
Moreover, in a MLMC framework, the white noise samples must be coupled between
subsequent levels. We show how our technique can be used to enforce this
coupling even in the case of non-nested mesh hierarchies. We demonstrate the
efficacy of our method with numerical experiments. We observe optimal
convergence rates for the finite element solution of the elliptic SPDEs of
interest in 2D and 3D and we show convergence of the sampled field covariances.
In a MLMC setting, a good coupling is enforced and the telescoping sum is
respected.Comment: 28 pages, 10 figure
Quantum phase transitions in effective spin-ladder models for graphene zigzag nanoribbons
We examine the magnetic correlations in quantum spin models that were derived
recently as effective low-energy theories for electronic correlation effects on
the edge states of graphene nanoribbons. For this purpose, we employ quantum
Monte Carlo simulations to access the large-distance properties, accounting for
quantum fluctuations beyond mean-field-theory approaches to edge magnetism. For
certain chiral nanoribbons, antiferromagnetic inter-edge couplings were
previously found to induce a gapped quantum disordered ground state of the
effective spin model. We find that the extended nature of the intra-edge
couplings in the effective spin model for zigzag nanoribbons leads to a quantum
phase transition at a large, finite value of the inter-edge coupling. This
quantum critical point separates the quantum disordered region from a gapless
phase of stable edge magnetism at weak intra-edge coupling, which includes the
ground states of spin-ladder models for wide zigzag nanoribbons. To study the
quantum critical behavior, the effective spin model can be related to a model
of two antiferromagnetically coupled Haldane-Shastry spin-half chains with
long-ranged ferromagnetic intra-chain couplings. The results for the critical
exponents are compared also to several recent renormalization group
calculations for related long-ranged interacting quantum systems.Comment: 12 pages, 15 figure
Hysteresis in Adiabatic Dynamical Systems: an Introduction
We give a nontechnical description of the behaviour of dynamical systems
governed by two distinct time scales. We discuss in particular memory effects,
such as bifurcation delay and hysteresis, and comment the scaling behaviour of
hysteresis cycles. These properties are illustrated on a few simple examples.Comment: 28 pages, 10 ps figures, AMS-LaTeX. This is the introduction of my
Ph.D. dissertation, available at
http://dpwww.epfl.ch/instituts/ipt/berglund/these.htm
Spacetime Approach to Phase Transitions
In these notes, the application of Feynman's sum-over-paths approach to
thermal phase transitions is discussed. The paradigm of such a spacetime
approach to critical phenomena is provided by the high-temperature expansion of
spin models. This expansion, known as the hopping expansion in the context of
lattice field theory, yields a geometric description of the phase transition in
these models, with the thermal critical exponents being determined by the
fractal structure of the high-temperature graphs. The graphs percolate at the
thermal critical point and can be studied using purely geometrical observables
known from percolation theory. Besides the phase transition in spin models and
in the closely related theory, other transitions discussed from this
perspective include Bose-Einstein condensation, and the transitions in the
Higgs model and the pure U(1) gauge theory.Comment: 59 pages, 18 figures. Write-up of Ising Lectures presented at the
National Academy of Sciences, Lviv, Ukraine, 2004. 2nd version: corrected
typo
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