We consider random walk and self-avoiding walk whose 1-step distribution is given by D, and oriented percolation whose bond-occupation probability is proportional to D. Suppose that D(x) decays as |x|−d − α with α > 0. For random walk in any dimension d and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension dc ≡ 2(α ∧ 2), we prove large-t asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length t or the average spatial size of an oriented percolation cluster at time t. This proves the conjecture for long-range self-avoiding walk in [Ann. Inst. H. Poincaré Probab. Statist. (2010), to appear] and for long-range oriented percolation in [Probab. Theory Related Fields 142 (2008) 151–188] and [Probab. Theory Related Fields 145 (2009) 435–458].
Publié le : 2011-03-15
Classification:
Long-range random walk,
self-avoiding walk,
oriented percolation,
gyration radius,
lace expansion,
60K35,
82B41,
82B43
@article{1298669172,
author = {Chen, Lung-Chi and Sakai, Akira},
title = {Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation},
journal = {Ann. Probab.},
volume = {39},
number = {1},
year = {2011},
pages = { 507-548},
language = {en},
url = {http://dml.mathdoc.fr/item/1298669172}
}
Chen, Lung-Chi; Sakai, Akira. Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation. Ann. Probab., Tome 39 (2011) no. 1, pp. 507-548. http://gdmltest.u-ga.fr/item/1298669172/