Random walk generated by random permutations of {1,2,3, ..., n+1}

TL;DR

This paper analyzes a permutation-generated random walk on a one-dimensional lattice, deriving exact probabilities, asymptotic normality with reduced diffusion, and characterizing excursions and turns, revealing novel non-Markovian properties.

Contribution

It provides an exact analysis of a non-Markovian permutation-based random walk, including probability distributions and asymptotic behavior, which is a novel contribution.

Findings

End-point distribution converges to a normal distribution with reduced diffusion coefficient.

Exact probability of the walk's end position is derived.

Asymptotic distribution of the number of turns is characterized.

Abstract

We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2, >...,n$, evolving in discrete time $l$ on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the \text{rise-and-descent} sequences characterizing random permutations $\pi$ of $[n+1] = \{1,2,3, ...,n+1\}$. We determine exactly the probability of finding the end-point $X_n = X^{(n)}_n$ of the trajectory of such a permutation-generated random walk (PGRW) at site $X$, and show that in the limit $n \to \infty$ it converges to a normal distribution with a smaller, compared to the conventional P\'olya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identic to the distribution of the intermediate points $X^{(n)}_l$, $l < n$, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of "turns" of the PGRW trajectories.