论文标题
审慎的步行尺寸六岁以上
Prudent walk in dimension six and higher
论文作者
论文摘要
我们研究了高维统一的谨慎自我避免步行，该步行将固定长度的所有最近邻邻自避免的自我避免的路径分配给审慎条件的所有近端自我避开路径，即，该路径不能朝着先前访问的位点的方向迈出任何步骤。我们证明，如果尺寸足够大，谨慎的自我避免行走会在扩散缩放下收敛到布朗运动。在尺寸d> 5的弱谨慎行事中，相同的结果也是如此。 高维审慎步行的挑战性特性是存在无限范围的自我避免约束。有趣的是，由于如此强大的自我避免限制，审慎步行的上限临界维度为五个，因此比经典的自我避免行走大。
We study the high-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. We prove that the prudent self-avoiding walk converges to Brownian motion under diffusive scaling if the dimension is large enough. The same result is true for weakly prudent walk in dimension d>5. A challenging property of the high-dimensional prudent walk is the presence of an infinite-range self-avoidance constraint. Interestingly, as a consequence of such a strong self-avoidance constraint, the upper critical dimension of the prudent walk is five, and thus greater than for the classical self-avoiding walk.
