论文标题
相关的随机步行和应用到记录的通用生存概率
Universal survival probability for a correlated random walk and applications to records
论文作者
论文摘要
我们考虑了一个空间连续的一维随机步行模型，并在步骤之间进行简单相关：两个连续步骤具有相同符号的概率为$ Q $，$ 0 \ leq q \ leq leq 1 $。因此，参数$ Q $允许控制随机步行的持久性。我们通过分析计算$ n $步行的步行概率，表明它与任何有限$ n $的跳跃分配无关。这种普遍性是Sparre-Andersen定理的结果，其随机步行不相关和对称步骤。然后，我们应用此结果来得出随机步行达到其最大值和步行记录统计的步骤的分布，该步行统计数据显示了相同的普遍性。特别是，我们表明，步行$ n \ gg 1 $ steps的记录数量的分布与$ n _ {\ rm eff}（q）（q）= n/（2（1-q））$不相关且对称分布的步骤相同。我们还表明，在$ n \ to \ infty $和$ q \ to $ y = n（1-q）$的制度中，该模型会收敛到运行式的粒子，这是一种持续的随机步行，通常用于模拟细菌的运动。我们的理论结果通过数值模拟证实。
We consider a model of space-continuous one-dimensional random walk with simple correlation between the steps: the probability that two consecutive steps have same sign is $q$ with $0\leq q\leq 1$. The parameter $q$ allows thus to control the persistence of the random walk. We compute analytically the survival probability of a walk of $n$ steps, showing that it is independent of the jump distribution for any finite $n$. This universality is a consequence of the Sparre-Andersen theorem for random walks with uncorrelated and symmetric steps. We then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk, which show the same universality. In particular, we show that the distribution of the number of records for a walk of $n\gg 1$ steps is the same as for a random walk with $n_{\rm eff}(q)=n/(2(1-q))$ uncorrelated and symmetrically distributed steps. We also show that in the regime where $n\to \infty$ and $q\to 1$ with $y=n(1-q)$, this model converges to the run-and-tumble particle, a persistent random walk often used to model the motion of bacteria. Our theoretical results are confirmed by numerical simulations.
