TITLE

# Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster

AUTHOR(S)
Lacoin, Hubert
PUB. DATE
August 2014
SOURCE
Probability Theory & Related Fields;Aug2014, Vol. 159 Issue 3/4, p777
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on $$\mathbb{Z }^d$$ . More precisely, we count $$Z_N$$ , the number of self-avoiding paths of length $$N$$ on the infinite cluster starting from the origin (which we condition to be in the cluster). We are interested in estimating the upper growth rate of $$Z_N$$ , $$\limsup _{N\rightarrow \infty } Z_N^{1/N}$$ , which 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 $$p\in (1/2,1)$$ , almost surely, $$Z_N$$ grows exponentially slower than its expected value. In other words, we prove that $$\limsup _{N\rightarrow \infty } (Z_N)^{1/N}{<}\lim _{N\rightarrow \infty } \mathbb{E }[Z_N]^{1/N}$$ , where the 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 walks on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on the specifics of percolation on $$\mathbb{Z }^2$$ , so our result can be extended to a large family of two-dimensional models including general self-avoiding walks in a random environment.
ACCESSION #
96985785

## Related Articles

• Efficient generation of low-energy folded states of a model protein. Gordon, Heather L.; Kwan, Wai Kei; Gong, Chunhang; Larrass, Stefan; Rothstein, Stuart M. // Journal of Chemical Physics;1/15/2003, Vol. 118 Issue 3, p1533

A number of short simulated annealing runs are performed on a highly-frustrated 46-"residue" off-lattice model protein. We perform, in an iterative fashion, a principal component analysis of the 946 nonbonded interbead distances, followed by two varieties of cluster analyses: hierarchical and...

• The number of infinite clusters in dynamical percolation. Peres, Yuval; Steif, Jeffrey E. // Probability Theory & Related Fields;1998, Vol. 111 Issue 1, p141

Summary. Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure as its stationary distribution. In previous work with O. Haggstrom, we found conditions for existence of infinite clusters at exceptional times. Here we show that...

• Incorporation of Effects of Diffusion into Advection-Mediated Dispersion in Porous Media. Hunt, A. G.; Skinner, T. E. // Journal of Statistical Physics;Aug2010, Vol. 140 Issue 3, p544

The distribution of solute arrival times, W( t; x), at position x in disordered porous media does not generally follow Gaussian statistics. A previous publication determined W( t; x) in the absence of diffusion from a synthesis of critical path, percolation scaling, and cluster statistics of...

• Phase Separation in Random Cluster Models III: Circuit Regularity. Hammond, Alan // Journal of Statistical Physics;Jan2011, Vol. 142 Issue 2, p229

We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Î“ encircling the origin and enclosing an area of at least (or exactly) n. In this paper, we prove that the resulting circuit is highly regular: we...

• A Particular Bit of Universality: Scaling Limits of Some Dependent Percolation Models. Camia, Federico; Newman, Charles M.; Sidoravicius, Vladas // Communications in Mathematical Physics;Apr2004, Vol. 246 Issue 2, p311

We study families of dependent site percolation models on the triangular lattice and hexagonal lattice H that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions...

• On the Number of Entangled Clusters. Atapour, Mahshid; Madras, Neal // Journal of Statistical Physics;Apr2010, Vol. 139 Issue 1, p1

We prove that the number of entangled clusters with N edges in the simple cubic lattice grows exponentially in N. This answers an open question posed by Grimmett and Holroyd (Proc. Lond. Math. Soc. 81:485ï¿½512, ). Our result has immediate implications for entanglement percolation: we obtain...

• Critical percolation: the expected number of clusters in a rectangle. Hongler, Clément; Smirnov, Stanislav // Probability Theory & Related Fields;Aug2011, Vol. 151 Issue 3/4, p735

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier...

• The Percolation Signature of the Spin Glass Transition. Machta, J.; Newman, C. M.; Stein, D. L. // Journal of Statistical Physics;Jan2008, Vol. 130 Issue 1, p113

Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuinâ€“Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin...

• Phase Separation in Random Cluster Models I: Uniform Upper Bounds on Local Deviation. Hammond, Alan // Communications in Mathematical Physics;Mar2012, Vol. 310 Issue 2, p455

This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the planar Fortuin-Kasteleyn random cluster model on the presence of an open dual circuit Î“ encircling the origin and enclosing an area of at...

Share