TY - JOUR
T1 - Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals
AU - Hara, Takashi
PY - 2008/3
Y1 - 2008/3
N2 - We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ℤd. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x € ℤd, the probability of a connection from the origin to x, and the generating functions for lattice trees or lattice animals containing the origin and x. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to const.|x| 2-d as |x| → ∞, for d ≥ 5 for self-avoiding walk, for d ≥ 19 for percolation, and for sufficiently large d for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349-408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if d > 4) condition under which the two-point function of a random walk on ℤd is asymptotic to const.|x|2-d as |x| → ∞.
AB - We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ℤd. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x € ℤd, the probability of a connection from the origin to x, and the generating functions for lattice trees or lattice animals containing the origin and x. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to const.|x| 2-d as |x| → ∞, for d ≥ 5 for self-avoiding walk, for d ≥ 19 for percolation, and for sufficiently large d for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349-408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if d > 4) condition under which the two-point function of a random walk on ℤd is asymptotic to const.|x|2-d as |x| → ∞.
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U2 - 10.1214/009117907000000231
DO - 10.1214/009117907000000231
M3 - Article
AN - SCOPUS:50249118414
VL - 36
SP - 530
EP - 593
JO - Annals of Probability
JF - Annals of Probability
SN - 0091-1798
IS - 2
ER -