## The Annals of Probability

### Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation

#### Abstract

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].

#### Article information

Source
Ann. Probab. Volume 39, Number 2 (2011), 507-548.

Dates
First available in Project Euclid: 25 February 2011

http://projecteuclid.org/euclid.aop/1298669172

Digital Object Identifier
doi:10.1214/10-AOP557

Mathematical Reviews number (MathSciNet)
MR2789505

Zentralblatt MATH identifier
1228.60108

#### Citation

Chen, Lung-Chi; Sakai, Akira. Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation. Ann. Probab. 39 (2011), no. 2, 507--548. doi:10.1214/10-AOP557. http://projecteuclid.org/euclid.aop/1298669172.

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