The ''true'' self-avoiding walk with bond repulsion on Z: Limit theorems

• 成果类型: Article
• 署名作者: Toth, B
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176987793
• 发表日期: 1995
• 页码: 1523-1556
• 关键词: reinforced random-walk

摘要

The ''true'' self-avoiding walk with bond repulsion is a nearest neighbor random walk on Z, for which the probability of jumping along a bond of the lattice is proportional to exp(-g . number of previous jumps along that bond). First we prove a limit theorem for the distribution of the local time process of this walk. Using this result, later we prove a local limit theorem, as A --> infinity, for the distribution of A(-2/3)X theta s/A, where theta s/A is a random time distributed geometrically with mean e(-s/A)(1-e(-s/A))-1 = A/s + O(1). As a by-product we also obtain an apparently new identity related to Brownian excursions and Bessel bridges.