674 research outputs found
Improved Bounds on the Phase Transition for the Hard-Core Model in 2-Dimensions
For the hard-core lattice gas model defined on independent sets weighted by
an activity , we study the critical activity
for the uniqueness/non-uniqueness threshold on the 2-dimensional integer
lattice . The conjectured value of the critical activity is
approximately . Until recently, the best lower bound followed from
algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating
the partition function for graphs of constant maximum degree when
where is the
infinite, regular tree of degree . His result established a certain
decay of correlations property called strong spatial mixing (SSM) on
by proving that SSM holds on its self-avoiding walk tree
where and is an ordering on the neighbors of vertex . As
a consequence he obtained that . Restrepo et al. (2011) improved Weitz's approach for
the particular case of and obtained that
. In this paper, we establish an upper bound for
this approach, by showing that, for all , SSM does not hold on
when . We also present a
refinement of the approach of Restrepo et al. which improves the lower bound to
.Comment: 19 pages, 1 figure. Polished proofs and examples compared to earlier
versio
Partially ordered models
We provide a formal definition and study the basic properties of partially
ordered chains (POC). These systems were proposed to model textures in image
processing and to represent independence relations between random variables in
statistics (in the later case they are known as Bayesian networks). Our chains
are a generalization of probabilistic cellular automata (PCA) and their theory
has features intermediate between that of discrete-time processes and the
theory of statistical mechanical lattice fields. Its proper definition is based
on the notion of partially ordered specification (POS), in close analogy to the
theory of Gibbs measure. This paper contains two types of results. First, we
present the basic elements of the general theory of POCs: basic geometrical
issues, definition in terms of conditional probability kernels, extremal
decomposition, extremality and triviality, reconstruction starting from
single-site kernels, relations between POM and Gibbs fields. Second, we prove
three uniqueness criteria that correspond to the criteria known as bounded
uniformity, Dobrushin and disagreement percolation in the theory of Gibbs
measures.Comment: 54 pages, 11 figures, 6 simulations. Submited to Journal of Stat.
Phy
Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles
In this paper we study a continuum version of the Potts model. Particles are
points in R^d, with a spin which may take S possible values, S being at least
3. Particles with different spins repel each other via a Kac pair potential. In
mean field, for any inverse temperature there is a value of the chemical
potential at which S+1 distinct phases coexist. For each mean field pure phase,
we introduce a restricted ensemble which is defined so that the empirical
particles densities are close to the mean field values. Then, in the spirit of
the Dobrushin Shlosman theory, we get uniqueness and exponential decay of
correlations when the range of the interaction is large enough. In a second
paper, we will use such a result to implement the Pirogov-Sinai scheme proving
coexistence of S+1 extremal DLR measures.Comment: 72 pages, 1 figur
The glueball spectrum from an anisotropic lattice study
The spectrum of glueballs below 4 GeV in the SU(3) pure-gauge theory is
investigated using Monte Carlo simulations of gluons on several anisotropic
lattices with spatial grid separations ranging from 0.1 to 0.4 fm. Systematic
errors from discretization and finite volume are studied, and the continuum
spin quantum numbers are identified. Care is taken to distinguish single
glueball states from two-glueball and torelon-pair states. Our determination of
the spectrum significantly improves upon previous Wilson action calculations.Comment: 14 pages, 8 figures, uses REVTeX and epsf.sty (final version
published in Physical Review D
Entropy and the variational principle for actions of sofic groups
Recently Lewis Bowen introduced a notion of entropy for measure-preserving
actions of a countable sofic group on a standard probability space admitting a
generating partition with finite entropy. By applying an operator algebra
perspective we develop a more general approach to sofic entropy which produces
both measure and topological dynamical invariants, and we establish the
variational principle in this context. In the case of residually finite groups
we use the variational principle to compute the topological entropy of
principal algebraic actions whose defining group ring element is invertible in
the full group C*-algebra.Comment: 44 pages; minor changes; to appear in Invent. Mat
Cosmological perturbations in Massive Gravity and the Higuchi bound
In de Sitter spacetime there exists an absolute minimum for the mass of a
spin-2 field set by the Higuchi bound m^2 \geq 2H^2. We generalize this bound
to arbitrary spatially flat FRW geometries in the context of the recently
proposed ghost-free models of Massive Gravity with an FRW reference metric, by
performing a Hamiltonian analysis for cosmological perturbations. We find that
the bound generically indicates that spatially flat FRW solutions in FRW
massive gravity, which exhibit a Vainshtein mechanism in the background as
required by consistency with observations, imply that the helicity zero mode is
a ghost. In contradistinction to previous works, the tension between the
Higuchi bound and the Vainshtein mechanism is equally strong regardless of the
equation of state for matter.Comment: 24 pages, typos and conventions correcte
Spectral properties of zero temperature dynamics in a model of a compacting granular column
The compacting of a column of grains has been studied using a one-dimensional
Ising model with long range directed interactions in which down and up spins
represent orientations of the grain having or not having an associated void.
When the column is not shaken (zero 'temperature') the motion becomes highly
constrained and under most circumstances we find that the generator of the
stochastic dynamics assumes an unusual form: many eigenvalues become
degenerate, but the associated multi-dimensional invariant spaces have but a
single eigenvector. There is no spectral expansion and a Jordan form must be
used. Many properties of the dynamics are established here analytically; some
are not. General issues associated with the Jordan form are also taken up.Comment: 34 pages, 4 figures, 3 table
Monte Carlo Methods for Estimating Interfacial Free Energies and Line Tensions
Excess contributions to the free energy due to interfaces occur for many
problems encountered in the statistical physics of condensed matter when
coexistence between different phases is possible (e.g. wetting phenomena,
nucleation, crystal growth, etc.). This article reviews two methods to estimate
both interfacial free energies and line tensions by Monte Carlo simulations of
simple models, (e.g. the Ising model, a symmetrical binary Lennard-Jones fluid
exhibiting a miscibility gap, and a simple Lennard-Jones fluid). One method is
based on thermodynamic integration. This method is useful to study flat and
inclined interfaces for Ising lattices, allowing also the estimation of line
tensions of three-phase contact lines, when the interfaces meet walls (where
"surface fields" may act). A generalization to off-lattice systems is described
as well.
The second method is based on the sampling of the order parameter
distribution of the system throughout the two-phase coexistence region of the
model. Both the interface free energies of flat interfaces and of (spherical or
cylindrical) droplets (or bubbles) can be estimated, including also systems
with walls, where sphere-cap shaped wall-attached droplets occur. The
curvature-dependence of the interfacial free energy is discussed, and estimates
for the line tensions are compared to results from the thermodynamic
integration method. Basic limitations of all these methods are critically
discussed, and an outlook on other approaches is given
Close-packed dimers on the line: diffraction versus dynamical spectrum
The translation action of \RR^{d} on a translation bounded measure
leads to an interesting class of dynamical systems, with a rather rich spectral
theory. In general, the diffraction spectrum of , which is the carrier
of the diffraction measure, live on a subset of the dynamical spectrum. It is
known that, under some mild assumptions, a pure point diffraction spectrum
implies a pure point dynamical spectrum (the opposite implication always being
true). For other systems, the diffraction spectrum can be a proper subset of
the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with
singular continuous diffraction) in \cite{EM}. Here, we construct a random
system of close-packed dimers on the line that have some underlying long-range
periodic order as well, and display the same type of phenomenon for a system
with absolutely continuous spectrum. An interpretation in terms of `atomic'
versus `molecular' spectrum suggests a way to come to a more general
correspondence between these two types of spectra.Comment: 14 pages, with some additions and improvement
Popular attitudes to memory, the body, and social identity : the rise of external commemoration in Britain, Ireland, and New England
A comparative analysis of samples of external memorials from burial grounds in Britain, Ireland and New England reveals a widespread pattern of change in monument style and content, and exponential growth in the number of permanent memorials from the 18th century onwards. Although manifested in regionally distinctive styles on which most academic attention has so far been directed, the expansion reflects global changes in social relationships and concepts of memory and the body. An archaeological perspective reveals the importance of external memorials in articulating these changing attitudes in a world of increasing material consumption
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