48,894 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
On the hard sphere model and sphere packings in high dimensions
We prove a lower bound on the entropy of sphere packings of of
density . The entropy measures how plentiful such
packings are, and our result is significantly stronger than the trivial lower
bound that can be obtained from the mere existence of a dense packing. Our
method also provides a new, statistical-physics-based proof of the lower bound on the maximum sphere packing density by showing
that the expected packing density of a random configuration from the hard
sphere model is at least when the
ratio of the fugacity parameter to the volume covered by a single sphere is at
least . Such a bound on the sphere packing density was first achieved
by Rogers, with subsequent improvements to the leading constant by Davenport
and Rogers, Ball, Vance, and Venkatesh
Birthday Inequalities, Repulsion, and Hard Spheres
We study a birthday inequality in random geometric graphs: the probability of
the empty graph is upper bounded by the product of the probabilities that each
edge is absent. We show the birthday inequality holds at low densities, but
does not hold in general. We give three different applications of the birthday
inequality in statistical physics and combinatorics: we prove lower bounds on
the free energy of the hard sphere model and upper bounds on the number of
independent sets and matchings of a given size in d-regular graphs.
The birthday inequality is implied by a repulsion inequality: the expected
volume of the union of spheres of radius r around n randomly placed centers
increases if we condition on the event that the centers are at pairwise
distance greater than r. Surprisingly we show that the repulsion inequality is
not true in general, and in particular that it fails in 24-dimensional
Euclidean space: conditioning on the pairwise repulsion of centers of
24-dimensional spheres can decrease the expected volume of their union
Spatial mixing and approximation algorithms for graphs with bounded connective constant
The hard core model in statistical physics is a probability distribution on
independent sets in a graph in which the weight of any independent set I is
proportional to lambda^(|I|), where lambda > 0 is the vertex activity. We show
that there is an intimate connection between the connective constant of a graph
and the phenomenon of strong spatial mixing (decay of correlations) for the
hard core model; specifically, we prove that the hard core model with vertex
activity lambda < lambda_c(Delta + 1) exhibits strong spatial mixing on any
graph of connective constant Delta, irrespective of its maximum degree, and
hence derive an FPTAS for the partition function of the hard core model on such
graphs. Here lambda_c(d) := d^d/(d-1)^(d+1) is the critical activity for the
uniqueness of the Gibbs measure of the hard core model on the infinite d-ary
tree. As an application, we show that the partition function can be efficiently
approximated with high probability on graphs drawn from the random graph model
G(n,d/n) for all lambda < e/d, even though the maximum degree of such graphs is
unbounded with high probability.
We also improve upon Weitz's bounds for strong spatial mixing on bounded
degree graphs (Weitz, 2006) by providing a computationally simple method which
uses known estimates of the connective constant of a lattice to obtain bounds
on the vertex activities lambda for which the hard core model on the lattice
exhibits strong spatial mixing. Using this framework, we improve upon these
bounds for several lattices including the Cartesian lattice in dimensions 3 and
higher.
Our techniques also allow us to relate the threshold for the uniqueness of
the Gibbs measure on a general tree to its branching factor (Lyons, 1989).Comment: 26 pages. In October 2014, this paper was superseded by
arxiv:1410.2595. Before that, an extended abstract of this paper appeared in
Proc. IEEE Symposium on the Foundations of Computer Science (FOCS), 2013, pp.
300-30
Disagreement percolation for the hard-sphere model
Disagreement percolation connects a Gibbs lattice gas and i.i.d. site
percolation on the same lattice such that non-percolation implies uniqueness of
the Gibbs measure. This work generalises disagreement percolation to the
hard-sphere model and the Boolean model. Non-percolation of the Boolean model
implies the uniqueness of the Gibbs measure and exponential decay of pair
correlations and finite volume errors. Hence, lower bounds on the critical
intensity for percolation of the Boolean model imply lower bounds on the
critical activity for a (potential) phase transition. These lower bounds
improve upon known bounds obtained by cluster expansion techniques. The proof
uses a novel dependent thinning from a Poisson point process to the hard-sphere
model, with the thinning probability related to a derivative of the free
energy
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
Scaling and Inverse Scaling in Anisotropic Bootstrap percolation
In bootstrap percolation it is known that the critical percolation threshold
tends to converge slowly to zero with increasing system size, or, inversely,
the critical size diverges fast when the percolation probability goes to zero.
To obtain higher-order terms (that is, sharp and sharper thresholds) for the
percolation threshold in general is a hard question. In the case of
two-dimensional anisotropic models, sometimes correction terms can be obtained
from inversion in a relatively simple manner.Comment: Contribution to the proceedings of the 2013 EURANDOM workshop
Probabilistic Cellular Automata: Theory, Applications and Future
Perspectives, equation typo corrected, constant of generalisation correcte
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