844 research outputs found
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
Non-coincidence of Quenched and Annealed Connective Constants on the supercritical planar percolation cluster
In this paper, we study the abundance of self-avoiding paths of a given
length on a supercritical percolation cluster on \bbZ^d. More precisely, we
count the number of self-avoiding paths of length on the infinite
cluster, starting from the origin (that we condition to be in the cluster). We
are interested in estimating the upper growth rate of , , that we call the connective constant of the dilute lattice.
After proving that this connective constant is a.s.\ non-random, we focus on
the two-dimensional case and show that for every percolation parameter , almost surely, grows exponentially slower than its expected
value. In other word we prove that \limsup_{N\to \infty} (Z_N)^{1/N}
<\lim_{N\to \infty} \bbE[Z_N]^{1/N} where expectation is taken with respect to
the percolation process. This result can be considered as a first mathematical
attempt to understand the influence of disorder for self-avoiding walk on a
(quenched) dilute lattice. Our method, which combines change of measure and
coarse graining arguments, does not rely on specifics of percolation on
\bbZ^2, so that our result can be extended to a large family of two
dimensional models including general self-avoiding walk in random environment.Comment: 25 pages. Version accepted for publication in PTR
New Lower Bounds on the Self-Avoiding-Walk Connective Constant
We give an elementary new method for obtaining rigorous lower bounds on the
connective constant for self-avoiding walks on the hypercubic lattice .
The method is based on loop erasure and restoration, and does not require exact
enumeration data. Our bounds are best for high , and in fact agree with the
first four terms of the expansion for the connective constant. The bounds
are the best to date for dimensions , but do not produce good results
in two dimensions. For , respectively, our lower bound is within
2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.Comment: 35 pages, 388480 bytes Postscript, NYU-TH-93/02/0
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
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