带有随机重置的Pólya游走返回原点
Returns to the origin of the Pólya walk with stochastic resetting
AI总结：
研究带随机重置的一维Pólya游走中自发返回与重置事件的联合统计，发现其累积量随时间线性增长且涨落服从大偏差函数，并揭示弱重置下的交叉现象及时间间隔的更新过程特性。
摘要：
我们考虑一维格点上的简单随机游走（或Pólya游走），其在每个时间步以概率$r$随机重置到原点。重点关注在观测时间$t$内，游走者自发返回原点的次数${\mathcal{N}}_t^{\times}$与重置事件次数${\mathcal{N}}_t^{\bullet}$的联合统计。这些次数在强意义上是时间广延的：它们的所有联合累积量均随$t$线性增长，且具有可显式计算的幅度，其涨落由平滑的双变量大偏差函数描述。在弱重置和长时间机制下，出现了非平凡的交叉现象。值得注意的是，重置随机游走自发返回原点的时间间隔构成了一个更新过程，该过程由单一的“修饰”概率分布描述。这些时间间隔是首个时间间隔（即“修饰”首达时间）的概率副本。本研究遵循了伴随论文中关于一般嵌套更新过程的更广泛研究。
英文摘要：
We consider the simple random walk (or Pólya walk) on the one-dimensional lattice subject to stochastic resetting to the origin with probability $r$ at each time step. The focus is on the joint statistics of the numbers ${\mathcal{N}}_t^{\times}$ of spontaneous returns of the walker to the origin and ${\mathcal{N}}_t^{\bullet}$ of resetting events up to some observation time $t$. These numbers are extensive in time in a strong sense: all their joint cumulants grow linearly in $t$, with explicitly computable amplitudes, and their fluctuations are described by a smooth bivariate large deviation function. A non-trivial crossover phenomenon takes place in the regime of weak resetting and late times. Remarkably, the time intervals between spontaneous returns to the origin of the reset random walk form a renewal process described in terms of a single `dressed' probability distribution. These time intervals are probabilistic copies of the first one, the `dressed' first-passage time. The present work follows a broader study, covered in a companion paper, on general nested renewal processes.
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