We consider an active run-and-tumble particle (RTP) in $d$ dimensions, starting from the origin and evolving over a time interval $[0,t]$. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are noninstantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the $x$ component does not change sign up to time $t$, showing that it does not depend on $d$. As a consequence of this result, we compute exactly other $x$-component properties, namely, the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e., they do not depend on $d$. Moreover, we show that these universal results hold also if the speed $v$ of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)].

中文翻译：

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任意尺寸滚动颗粒的通用性质

我们考虑一个活跃的滚滚粒子（RTP） $d$ 尺寸，从原点开始并随时间间隔发展 $[0，Ť]$。我们检查了RTP动力学的三种不同模型：具有瞬时滚动的标准RTP模型，具有瞬时运行的变体以及其中滚动和运行均非瞬时的通用模型。对于这些模型中的每一个，我们将Sparre Andersen定理用于离散时间随机游动，以准确计算出$X$ 组件不会更改注册时间 $Ť$，表明它不依赖于 $d$。作为此结果的结果，我们精确计算了其他$X$-组件属性，即最大时间的分布和记录统计信息，表明它们是通用的，即它们不依赖于 $d$。而且，我们证明，如果速度$v$从一般概率分布中得出，每次翻滚后粒子的“随机数”都是随机的。我们的发现被数值模拟所证实。其中一些结果已在最近的一封信中宣布。牧师 124，090603（2020）]。