4,526 research outputs found
On the approach to equilibrium for a polymer with adsorption and repulsion
We consider paths of a one-dimensional simple random walk conditioned to come
back to the origin after L steps (L an even integer). In the 'pinning model'
each path \eta has a weight \lambda^{N(\eta)}, where \lambda>0 and N(\eta) is
the number of zeros in \eta. When the paths are constrained to be non-negative,
the polymer is said to satisfy a hard-wall constraint. Such models are well
known to undergo a localization/delocalization transition as the pinning
strength \lambda is varied. In this paper we study a natural 'spin flip'
dynamics for these models and derive several estimates on its spectral gap and
mixing time. In particular, for the system with the wall we prove that
relaxation to equilibrium is always at least as fast as in the free case
(\lambda=1, no wall), where the gap and the mixing time are known to scale as
L^{-2} and L^2\log L, respectively. This improves considerably over previously
known results. For the system without the wall we show that the equilibrium
phase transition has a clear dynamical manifestation: for \lambda \geq 1 the
relaxation is again at least as fast as the diffusive free case, but in the
strictly delocalized phase (\lambda < 1) the gap is shown to be O(L^{-5/2}), up
to logarithmic corrections. As an application of our bounds, we prove stretched
exponential relaxation of local functions in the localized regime.Comment: 43 pages, 5 figures; v2: corrected typos, added Table
A one-dimensional coagulation-fragmentation process with a dynamical phase transition
We introduce a reversible Markovian coagulation-fragmentation process on the
set of partitions of into disjoint intervals. Each interval
can either split or merge with one of its two neighbors. The invariant measure
can be seen as the Gibbs measure for a homogeneous pinning model
\cite{cf:GBbook}. Depending on a parameter , the typical configuration
can be either dominated by a single big interval (delocalized phase), or be
composed of many intervals of order (localized phase), or the interval
length can have a power law distribution (critical regime). In the three cases,
the time required to approach equilibrium (in total variation) scales very
differently with . In the localized phase, when the initial condition is a
single interval of size , the equilibration mechanism is due to the
propagation of two "fragmentation fronts" which start from the two boundaries
and proceed by power-law jumps
High-order conservative finite difference GLM-MHD schemes for cell-centered MHD
We present and compare third- as well as fifth-order accurate finite
difference schemes for the numerical solution of the compressible ideal MHD
equations in multiple spatial dimensions. The selected methods lean on four
different reconstruction techniques based on recently improved versions of the
weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving
(MP) schemes as well as slope-limited polynomial reconstruction. The proposed
numerical methods are highly accurate in smooth regions of the flow, avoid loss
of accuracy in proximity of smooth extrema and provide sharp non-oscillatory
transitions at discontinuities. We suggest a numerical formulation based on a
cell-centered approach where all of the primary flow variables are discretized
at the zone center. The divergence-free condition is enforced by augmenting the
MHD equations with a generalized Lagrange multiplier yielding a mixed
hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175
(2002) 645-673). The resulting family of schemes is robust, cost-effective and
straightforward to implement. Compared to previous existing approaches, it
completely avoids the CPU intensive workload associated with an elliptic
divergence cleaning step and the additional complexities required by staggered
mesh algorithms. Extensive numerical testing demonstrate the robustness and
reliability of the proposed framework for computations involving both smooth
and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics
(Aug 7 2009
From Competition to Complementarity: Comparative Influence Diffusion and Maximization
Influence maximization is a well-studied problem that asks for a small set of
influential users from a social network, such that by targeting them as early
adopters, the expected total adoption through influence cascades over the
network is maximized. However, almost all prior work focuses on cascades of a
single propagating entity or purely-competitive entities. In this work, we
propose the Comparative Independent Cascade (Com-IC) model that covers the full
spectrum of entity interactions from competition to complementarity. In Com-IC,
users' adoption decisions depend not only on edge-level information
propagation, but also on a node-level automaton whose behavior is governed by a
set of model parameters, enabling our model to capture not only competition,
but also complementarity, to any possible degree. We study two natural
optimization problems, Self Influence Maximization and Complementary Influence
Maximization, in a novel setting with complementary entities. Both problems are
NP-hard, and we devise efficient and effective approximation algorithms via
non-trivial techniques based on reverse-reachable sets and a novel "sandwich
approximation". The applicability of both techniques extends beyond our model
and problems. Our experiments show that the proposed algorithms consistently
outperform intuitive baselines in four real-world social networks, often by a
significant margin. In addition, we learn model parameters from real user
action logs.Comment: An abridged of this work is to appear in the Proceedings of VLDB
Endowment (PVDLB), Vol 9, No 2. Also, the paper will be presented in the VLDB
2016 conference in New Delhi, India. This update contains new theoretical and
experimental results, and the paper is now in single-column format (44 pages
Glauber dynamics for the quantum Ising model in a transverse field on a regular tree
Motivated by a recent use of Glauber dynamics for Monte-Carlo simulations of
path integral representation of quantum spin models [Krzakala, Rosso,
Semerjian, and Zamponi, Phys. Rev. B (2008)], we analyse a natural Glauber
dynamics for the quantum Ising model with a transverse field on a finite graph
. We establish strict monotonicity properties of the equilibrium
distribution and we extend (and improve) the censoring inequality of Peres and
Winkler to the quantum setting. Then we consider the case when is a regular
-ary tree and prove the same fast mixing results established in [Martinelli,
Sinclair, and Weitz, Comm. Math. Phys. (2004)] for the classical Ising model.
Our main tool is an inductive relation between conditional marginals (known as
the "cavity equation") together with sharp bounds on the operator norm of the
derivative at the stable fixed point. It is here that the main difference
between the quantum and the classical case appear, as the cavity equation is
formulated here in an infinite dimensional vector space, whereas in the
classical case marginals belong to a one-dimensional space
The random geometry of equilibrium phases
This is a (long) survey about applications of percolation theory in
equilibrium statistical mechanics. The chapters are as follows:
1. Introduction
2. Equilibrium phases
3. Some models
4. Coupling and stochastic domination
5. Percolation
6. Random-cluster representations
7. Uniqueness and exponential mixing from non-percolation
8. Phase transition and percolation
9. Random interactions
10. Continuum modelsComment: 118 pages. Addresses: [email protected]
http://www.mathematik.uni-muenchen.de/~georgii.html [email protected]
http://www.math.chalmers.se/~olleh [email protected]
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