Godrèche Claude, Majumdar Satya N, Schehr Grégory
Institut de Physique Théorique, Université Paris-Saclay, CEA and CNRS, 91191 Gif-sur-Yvette, France.
LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.
Phys Rev Lett. 2016 Jul 1;117(1):010601. doi: 10.1103/PhysRevLett.117.010601. Epub 2016 Jun 29.
We study the statistics of increments in record values in a time series {x_{0}=0,x_{1},x_{2},…,x_{n}} generated by the positions of a random walk (discrete time, continuous space) of duration n steps. For arbitrary jump length distribution, including Lévy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of n for large n, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability Q(n) that the record increments decrease monotonically up to step n. Remarkably, Q(n) is universal (i.e., independent of the jump distribution) for each n, decaying as Q(n)∼A/sqrt[n] for large n, with a universal amplitude A=e/sqrt[π]=1.53362….
我们研究了由持续时间为(n)步的随机游走（离散时间，连续空间）位置生成的时间序列({x_{0}=0,x_{1},x_{2},\cdots,x_{n}})中记录值增量的统计特性。对于任意的跳跃长度分布，包括 Lévy 飞行，我们表明记录增量的分布变得平稳，即对于大的(n)，与(n)无关，并针对广泛的跳跃分布类别明确计算了它。此外，我们精确计算了记录增量在直到第(n)步单调递减的概率(Q(n))。值得注意的是，对于每个(n)，(Q(n))是通用的（即与跳跃分布无关），对于大的(n)，(Q(n))按(Q(n)\sim A/\sqrt{n})衰减，通用幅度(A = e/\sqrt{\pi}=1.53362\cdots) 。
