论文标题
重置随机助行器访问的不同站点数量
Number of distinct sites visited by a resetting random walker
论文作者
论文摘要
我们研究了$ n $ step重置随机助行器在$ d $ d $ d $ d $二维的超纤维晶格上访问的不同站点的数字$ v_p（n）$，带有重置概率$ p $。在情况下，$ p = 0 $，我们恢复了众所周知的结果，即大$ n $的平均数量增加了$ \ langle v_0（n）\ rangle \ sim n^{d/2} $，$ d <2 $和$ \ langle v_0（n）\ langle v_0（n）\ rangle \ sim n $ for $ d> $ d $ d $ d> 2 $。对于$ p> 0 $，我们表明$ \ langle v_p（n）\ rangle $生长得非常慢，因为$ \ sim \ left [\ log（n）\ right]^d $。我们观察到，在重置存在的情况下，标准随机步行（无重置）的复发 - 转变过渡（无重置）消失了。在限制$ p \至0 $中，我们计算两个制度之间的精确交叉缩放函数。在一维情况下，我们通过分析得出$ n $限制的$ v_p（n）$的完整分布。此外，对于一维随机助行器，我们引入了一种新的可观察到的可观察到的，我们称之为不平衡，该沃克量衡量了访问的区域在起始位置周围对称的程度。我们通过分析计算以$ p = 0 $和$ p> 0 $的全面分配。我们的理论结果通过广泛的数值模拟得到验证。
We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step resetting random walker on a $d$-dimensional hypercubic lattice with resetting probability $p$. In the case $p=0$, we recover the well-known result that the average number of distinct sites grows for large $n$ as $\langle V_0(n)\rangle\sim n^{d/2}$ for $d<2$ and as $\langle V_0(n)\rangle\sim n$ for $d>2$. For $p>0$, we show that $\langle V_p(n)\rangle$ grows extremely slowly as $\sim \left[\log(n)\right]^d$. We observe that the recurrence-transience transition at $d=2$ for standard random walks (without resetting) disappears in the presence of resetting. In the limit $p\to 0$, we compute the exact crossover scaling function between the two regimes. In the one-dimensional case, we derive analytically the full distribution of $V_p(n)$ in the limit of large $n$. Moreover, for a one-dimensional random walker, we introduce a new observable, which we call imbalance, that measures how much the visited region is symmetric around the starting position. We analytically compute the full distribution of the imbalance both for $p=0$ and for $p>0$. Our theoretical results are verified by extensive numerical simulations.
