Random patterns generated by random permutations of natural numbers

TL;DR

This paper surveys recent findings on patterns generated by random permutations of natural numbers, analyzing random walks and surface properties on 1D and 2D lattices, revealing probabilistic behaviors and collective phenomena.

Contribution

It provides a comprehensive analysis of permutation-induced patterns, including exact probabilities and distribution functions for random walks and surface peaks, highlighting new statistical properties.

Findings

Exact probability of permutation-based random walk positions

Distribution of excursions and U-turns in the walk

Distribution of local peaks on 1D and 2D lattices

Abstract

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time $n$, whose moves to the right or to the left are induced by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site $X$ at time $n$, obtain the probability measure of different excursions and define the asymptotic distribution of the number of "U-turns" of the trajectories - permutation "peaks" and "through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers $1,2,3, >...,L$ on sites of a 1d or 2d square lattices containing $L$ sites. We calculate the distribution function of the number of local "peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss some surprising collective behavior emerging in this model.