Universal distribution of the number of minima for random walks and Lévy flights

Abstract

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 in the latter $m^{-3/2}$ 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.

arXiv commentTypos corrected, submitted version. Main text: 6 pages, 3 figures. Supplementary material: 20 pages, 12 figures

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Cited by in corpus (1)

• The number of minima in random landscapes generated by constrained random walk and Lévy flights: universal properties