Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation
نویسندگان
چکیده
منابع مشابه
Long-range self-avoiding walk converges to α-stable processes
Abstract: We consider a long-range version of self-avoiding walk in dimension d > 2(α ∧ 2), where d denotes dimension and α the power-law decay exponent of the coupling function. Under appropriate scaling we prove convergence to Brownian motion for α ≥ 2, and to α-stable Lévy motion for α < 2. This complements results by Slade (1988), who proves convergence to Brownian motion for nearest-neighb...
متن کاملTransience, Recurrence and Critical Behavior for Long-Range Percolation
We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions d = 1, 2, where x and y are connected with probability ∼ β/‖x − y‖−s. We show that when d < s < 2d the walk is transient, and when s ≥ 2d, the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we ge...
متن کاملCritical behavior and the limit distribution for long - range oriented percolation
We consider oriented percolation on Z×Z+ whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on Zd. Suppose that D(x) decays as |x|−d−α for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective meanfield values above the upper-critical dime...
متن کاملCritical behavior and the limit distribution for long - range oriented percolation . II : Spatial correlation
We prove that the Fourier transform of the properly-scaled normalized twopoint function for sufficiently spread-out long-range oriented percolation with index α > 0 converges to e−C|k| α∧2 for some C ∈ (0,∞) above the upper-critical dimension dc ≡ 2(α ∧ 2). This answers the open question remained in the previous paper [1]. Moreover, we show that the constant C exhibits crossover at α = 2, which...
متن کاملLong-range percolation in R
Let X be either Z or the points of a Poisson process in R of intensity 1. Given parameters r and p, join each pair of points of X within distance r independently with probability p. This is the simplest case of a ‘spreadout’ percolation model studied by Penrose [6], who showed that, as r → ∞, the average degree of the corresponding random graph at the percolation threshold tends to 1, i.e., the...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: The Annals of Probability
سال: 2011
ISSN: 0091-1798
DOI: 10.1214/10-aop557