48,161 research outputs found
Random-Cluster Dynamics in
The random-cluster model has been widely studied as a unifying framework for
random graphs, spin systems and electrical networks, but its dynamics have so
far largely resisted analysis. In this paper we analyze the Glauber dynamics of
the random-cluster model in the canonical case where the underlying graph is an
box in the Cartesian lattice . Our main result is a
upper bound for the mixing time at all values of the model
parameter except the critical point , and for all values of the
second model parameter . We also provide a matching lower bound proving
that our result is tight. Our analysis takes as its starting point the recent
breakthrough by Beffara and Duminil-Copin on the location of the random-cluster
phase transition in . It is reminiscent of similar results for
spin systems such as the Ising and Potts models, but requires the reworking of
several standard tools in the context of the random-cluster model, which is not
a spin system in the usual sense
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
Matching measure, Benjamini-Schramm convergence and the monomer-dimer free energy
We define the matching measure of a lattice L as the spectral measure of the
tree of self-avoiding walks in L. We connect this invariant to the
monomer-dimer partition function of a sequence of finite graphs converging to
L.
This allows us to express the monomer-dimer free energy of L in terms of the
measure. Exploiting an analytic advantage of the matching measure over the
Mayer series then leads to new, rigorous bounds on the monomer-dimer free
energies of various Euclidean lattices. While our estimates use only the
computational data given in previous papers, they improve the known bounds
significantly.Comment: 18 pages, 3 figure
The lattice dimension of a graph
We describe a polynomial time algorithm for, given an undirected graph G,
finding the minimum dimension d such that G may be isometrically embedded into
the d-dimensional integer lattice Z^d.Comment: 6 pages, 3 figure
Counting hypergraph matchings up to uniqueness threshold
We study the problem of approximately counting matchings in hypergraphs of
bounded maximum degree and maximum size of hyperedges. With an activity
parameter , each matching is assigned a weight .
The counting problem is formulated as computing a partition function that gives
the sum of the weights of all matchings in a hypergraph. This problem unifies
two extensively studied statistical physics models in approximate counting: the
hardcore model (graph independent sets) and the monomer-dimer model (graph
matchings).
For this model, the critical activity
is the threshold for the uniqueness of Gibbs measures on the infinite
-uniform -regular hypertree. Consider hypergraphs of maximum
degree at most and maximum size of hyperedges at most . We show that
when , there is an FPTAS for computing the partition
function; and when , there is a PTAS for computing the
log-partition function. These algorithms are based on the decay of correlation
(strong spatial mixing) property of Gibbs distributions. When , there is no PRAS for the partition function or the log-partition
function unless NPRP.
Towards obtaining a sharp transition of computational complexity of
approximate counting, we study the local convergence from a sequence of finite
hypergraphs to the infinite lattice with specified symmetry. We show a
surprising connection between the local convergence and the reversibility of a
natural random walk. This leads us to a barrier for the hardness result: The
non-uniqueness of infinite Gibbs measure is not realizable by any finite
gadgets
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