Kundu Anupam, Majumdar Satya N, Schehr Grégory
<a href="https://ror.org/0015qa126">International Centre for Theoretical Sciences</a>, <a href="https://ror.org/03ht1xw27">Tata Institute of Fundamental Research</a>, Bengaluru-560089, India.
<a href="https://ror.org/00w67e447">LPTMS</a>, CNRS, Université Paris-Sud, <a href="https://ror.org/03xjwb503">Université Paris-Saclay</a>, 91405 Orsay, France.
Phys Rev E. 2024 Aug;110(2-1):024137. doi: 10.1103/PhysRevE.110.024137.
We compute exactly the full distribution of the number m of local minima in a one-dimensional landscape generated by a random walk or a Lévy flight. We consider two different ensembles of landscapes, one with a fixed number of steps N and the other till the first-passage time of the random walk to the origin. We show that the distribution of m is drastically different in the two ensembles (Gaussian in the former case, while having a power-law tail m^{-3/2} in the latter case). However, the most striking aspect of our results is that, in each case, the distribution is completely universal for all m (and not just for large m), i.e., independent of the jump distribution in the random walk. This means that the distributions are exactly identical for Lévy flights and random walks with finite jump variance. Our analytical results are in excellent agreement with our numerical simulations.
我们精确计算了由随机游走或 Lévy 飞行生成的一维景观中局部极小值数量 m 的完整分布。我们考虑了两种不同的景观系综，一种是具有固定步数 N 的系综，另一种是直到随机游走首次回到原点的时间的系综。我们表明，在这两种系综中 m 的分布截然不同（在前一种情况下是高斯分布，而在后一种情况下具有幂律尾部 m^(-3/2)）。然而，我们结果最引人注目的方面是，在每种情况下，对于所有 m（而不仅仅是大 m）分布都是完全通用的，即与随机游走中的跳跃分布无关。这意味着对于具有有限跳跃方差的 Lévy 飞行和随机游走，分布完全相同。我们的分析结果与数值模拟结果非常吻合。
