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• W2065009430 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 $$Z_N$$ , $$limsup _{Nrightarrow infty } Z_N^{1/N}$$ , 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 $$pin (1/2,1)$$ , almost surely, $$Z_N$$ grows exponentially slower than its expected value. In other words, we prove that $$limsup _{Nrightarrow infty } (Z_N)^{1/N}{<}lim _{Nrightarrow infty } mathbb{E }[Z_N]^{1/N}$$ , 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." @default.
• W2065009430 created "2016-06-24" @default.
• W2065009430 creator A5028955238 @default.
• W2065009430 date "2013-08-13" @default.
• W2065009430 modified "2023-10-05" @default.
• W2065009430 title "Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster" @default.
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• W2065009430 doi "https://doi.org/10.1007/s00440-013-0520-1" @default.
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