The number of minima in random landscapes generated by constrained random walk and L\'evy flights: universal properties

TL;DR

This paper develops a unified framework to exactly compute the distribution of the number of minima in one-dimensional random walk landscapes generated by various constrained and unconstrained stochastic processes, revealing universal properties independent of jump distribution.

Contribution

It introduces a universal analytical approach for different constrained random walk models, linking minima distribution to the Sparre Andersen theorem, and confirms results with numerical simulations.

Findings

Distribution of minima is universal across models for symmetric continuous jumps.

Mapping to Sparre Andersen theorem explains universality in constrained models.

Analytical results match numerical simulations perfectly.

Abstract

We provide a uniform framework to compute the exact distribution of the number of minima/maxima in three different random walk landscape models in one dimension. The landscape is generated by the trajectory of a discrete-time continuous space random walk with arbitrary symmetric and continuous jump distribution at each step. In model I, we consider a ``free'' random walk of $N$ steps. In model II, we consider a ``meander landscape'' where the random walk, starting at the origin, stays non-negative up to $N$ steps. In model III, we study a ``first-passage landscape'' which is generated by the trajectory of a random walk that starts at the origin and stops when it crosses the origin for the first time. We demonstrate that while the exact distribution of the number of minima is different in the three models, for each model it is universal for all $N$, in the sense that it does not depend on the jump distribution as long as it is symmetric and continuous. In the last two cases we show that this universality follows from a non trivial mapping to the Sparre Andersen theorem known for the first-passage probability of discrete-time random walks with symmetric and continuous jump distribution. Our analytical results are in excellent agreement with our numerical simulations.