38 research outputs found
Uniform Lipschitz functions on the triangular lattice have logarithmic variations
Uniform integer-valued Lipschitz functions on a domain of size of the
triangular lattice are shown to have variations of order . The
level lines of such functions form a loop model on the edges of the
hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for
the loop O(2) model is constructed as a thermodynamic limit and is shown to be
unique. It contains only finite loops and has properties indicative of
scale-invariance: macroscopic loops appearing at every scale. The existence of
the infinite-volume measure carries over to height functions pinned at the
origin; the uniqueness of the Gibbs measure does not. The proof is based on a
representation of the loop model via a pair of spin configurations that
are shown to satisfy the FKG inequality. We prove RSW-type estimates for a
certain connectivity notion in the aforementioned spin model.Comment: Compared to v1: Theorem 1.3 (uniqueness of the Gibbs measure) is
added; proof of Theorem 3.2 (delocalization) is significantly shortened; more
details added in Section 4 (proof of the dichotomy theorem
On the probability that self-avoiding walk ends at a given point
We prove two results on the delocalization of the endpoint of a uniform
self-avoiding walk on Z^d for d>1. We show that the probability that a walk of
length n ends at a point x tends to 0 as n tends to infinity, uniformly in x.
Also, for any fixed x in Z^d, this probability decreases faster than n^{-1/4 +
epsilon} for any epsilon >0. When |x|= 1, we thus obtain a bound on the
probability that self-avoiding walk is a polygon.Comment: 31 pages, 8 figures. Referee corrections implemented; removed section
5.
Inhomogeneous bond percolation on square, triangular and hexagonal lattices
The star-triangle transformation is used to obtain an equivalence extending
over the set of all (in)homogeneous bond percolation models on the square,
triangular and hexagonal lattices. Among the consequences are box-crossing
(RSW) inequalities for such models with parameter-values at which the
transformation is valid. This is a step toward proving the universality and
conformality of these processes. It implies criticality of such values, thereby
providing a new proof of the critical point of inhomogeneous systems. The
proofs extend to certain isoradial models to which previous methods do not
apply.Comment: Published in at http://dx.doi.org/10.1214/11-AOP729 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org