We study the statistical properties of the convex hull of a planar run-and-tumble particle (RTP), also known as the "persistent random walk", where the particle/walker runs ballistically between tumble events at which it changes its direction randomly. We consider two different statistical ensembles where we either fix (i) the total number of tumblings $n$ or (ii) the total duration $t$ of the time interval. In both cases, we derive exact expressions for the average perimeter of the convex hull and then compare to numerical estimates finding excellent agreement. Further, we numerically compute the full distribution of the perimeter using Markov chain Monte Carlo techniques, in both ensembles, probing the far tails of the distribution, up to a precision smaller than $10^{-100}$. This also allows us to characterize the rare events that contribute to the tails of these distributions.

中文翻译：

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平面中奔跑和翻滚粒子的凸包

我们研究了平面奔跑和翻滚粒子 (RTP) 的凸包的统计特性，也称为“持久随机游走”，其中粒子/步行者在翻滚事件之间弹道运行，并随机改变其方向。我们考虑两种不同的统计集合，其中我们确定 (i) 翻滚的总数 $n$ 或 (ii) 时间间隔的总持续时间 $t$。在这两种情况下，我们推导出凸包平均周长的精确表达式，然后与数值估计进行比较，发现非常一致。此外，我们使用马尔可夫链蒙特卡罗技术在数值上计算周长的完整分布，在两个集成中探测分布的远尾，精度小于 $10^{-100}$。