论文标题
$ {1D} $ lattice的随机步行和Lévy航班的记录数量的统计数据
Statistics of the Number of Records for Random Walks and Lévy Flights on a ${1D}$ Lattice
论文作者
论文摘要
我们研究对称的记录$ r_n $的统计数据，$ n $ step，$ n $ step，$ n $ step，$ n $ step，$ n $ step，$ n $ step，$ n $ step，$ n $ step，$ n $ step $ n $ step $ n $ step $ n $ step。在给定的步骤中，步行者可以从给定的对称概率分布中提取的任意晶格单元跳跃。此过程包括特殊情况，是标准最近的邻居晶格随机步行。我们明确得出记录数量的分布$ p（r_n）的生成功能，对任意离散跳跃分布有效。作为副产品，我们提供了一个相对简单的证据，证明了广义的Andersen定理，用于在线上随机行走，具有离散或连续的跳跃分布的生存概率。对于离散的跳跃过程，我们将得出$ p（r_n）$的渐近大$ n $行为以及记录$ e（r_n）$的平均数量。我们表明，与随机步行的情况不同，具有对称和连续的跳跃分布，其中记录统计数据非常通用（即，与所有$ n $的跳跃分布无关），晶格步行的记录统计信息取决于任何固定$ n $的跳跃分布。但是，在大$ n $限制中，我们表明缩放记录号的分布$ r_n/e（r_n）$接近任何离散跳跃过程的通用，半高斯的形式。对跳跃分布的依赖性仅通过比例因子$ e（r_n）$进入，我们还以任意跳跃分布的大$ n $限制计算。我们提供了一些示例的明确结果，并提供了我们的分析预测的数值检查。
We study the statistics of the number of records $R_n$ for a symmetric, $n$-step, discrete jump process on a $1D$ lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution $P(R_n)$ of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival probability of a random walk on a line, with discrete or continuous jump distributions. For the discrete jump process, we then derive the asymptotic large $n$ behavior of $P(R_n)$ as well as of the average number of records $E(R_n)$. We show that unlike the case of random walks with symmetric and continuous jump distributions where the record statistics is strongly universal (i.e., independent of the jump distribution for all $n$), the record statistics for lattice walks depends on the jump distribution for any fixed $n$. However, in the large $n$ limit, we show that the distribution of the scaled record number $R_n/E(R_n)$ approaches a universal, half-Gaussian form for any discrete jump process. The dependence on the jump distribution enters only through the scale factor $E(R_n)$, which we also compute in the large $n$ limit for arbitrary jump distributions. We present explicit results for a few examples and provide numerical checks of our analytical predictions.
