论文标题
缩放限制的随机步行限制在维度一中受到其范围的惩罚
Scaling limits for the random walk penalized by its range in dimension one
论文作者
论文摘要
在本文中，我们研究了一个差溶剂中聚合物的一维模型：$ \ mathbb {z} $在其范围内进行了惩罚。更确切地说，我们考虑了gibbs对简单的符号随机步行的定律转换，重量$ \ exp（-h_n | r_n |）$，带有$ | r_n | $ $ | r_n | $访问的站点的数量和$ h_n $ a size size依赖依赖的正极参数。我们使用赌徒的废墟估计来获得分区功能的精确渐近学，这使我们能够获得轨迹的精确描述，特别是对中心的缩放限制和范围的幅度。在$ h_n \大约n^{1/4} $上识别出最佳幅度波动的相变，该晶格结构固有。
In this article we study a one dimensional model for a polymer in a poor solvent: the random walk on $\mathbb{Z}$ penalized by its range. More precisely, we consider a Gibbs transformation of the law of the simple symmmetric random walk by a weight $\exp(-h_n|R_n|)$, with $|R_n|$ the number of visited sites and $h_n$ a size-dependent positive parameter. We use gambler's ruin estimates to obtain exact asymptotics for the partition function, that enables us to obtain a precise description of trajectories, in particular scaling limits for the center and the amplitude of the range. A phase transition for the fluctuations around an optimal amplitude is identified at $h_n \approx n^{1/4}$ , inherent to the underlying lattice structure.
