The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions
نویسندگان
چکیده
We prove existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on Z d . For the critical point, de ned to be the reciprocal of the connective constant, the coe cients of the expansion are computed through order d 6 , with a rigorous error bound of order d 7 . Our method for computing terms in the expansion also applies to percolation, and for nearest-neighbour independent Bernoulli bond percolation on Z d gives the 1=d-expansion for the critical point through order d 3 , with a rigorous error bound of order d 4 . The method uses the lace expansion.
منابع مشابه
Critical points for spread - out self - avoiding walk , percolation and the contact process above the upper critical dimensions Remco
We consider self-avoiding walk and percolation in Zd, oriented percolation in Z×Z+, and the contact process in Zd, with pD( · ) being the coupling function whose range is denoted by L < ∞. For percolation, for example, each bond {x, y} is occupied with probability pD(y−x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical poin...
متن کاملCritical points for spread - out self - avoiding walk , percolation and the contact process above the upper critical dimensions
We consider self-avoiding walk and percolation in Zd, oriented percolation in Z×Z+, and the contact process in Zd, with p D( · ) being the coupling function whose range is denoted by L < ∞. For percolation, for example, each bond {x, y} is occupied with probability p D(y−x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical po...
متن کاملCritical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models
We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on Zd, having long finite-range connections, above their upper critical dimensions d = 4 (self-avoiding walk), d = 6 (percolation) and d = 8 (trees and animals). The two-point functions for these models are respectively the generating function for selfavoiding walks from the origin to x...
متن کاملCritical Two-point Functions and the Lace Expansion for Spread-out High-dimensional Percolation and Related Models by Takashi Hara,1 Remco
We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on Zd , having long finite-range connections, above their upper critical dimensions d = 4 (self-avoiding walk), d = 6 (percolation) and d = 8 (trees and animals). The two-point functions for these models are respectively the generating function for selfavoiding walks from the origin to ...
متن کاملThe lace expansion on a tree with application to networks of self-avoiding walks
The lace expansion has been used successfully to study the critical behaviour in high dimensions of self-avoiding walks, lattice trees and lattice animals, and percolation. In each case, the lace expansion has been an expansion along a time interval. In this paper, we introduce the lace expansion on a tree, in which ‘time’ is generalised from an interval to a tree. We develop the expansion in t...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Combinatorics, Probability & Computing
دوره 4 شماره
صفحات -
تاریخ انتشار 1995