## The Annals of Probability

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

### 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 *d*_{c} ≡ 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/10-AOP557

**Mathematical Reviews number (MathSciNet)**

MR2789505

**Zentralblatt MATH identifier**

05878715

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B43: Percolation [See also 60K35]

**Keywords**

Long-range random walk self-avoiding walk oriented percolation gyration radius lace expansion

#### 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.