Elephants explore in spirals sometimes

TL;DR

This work analyzes Elephant Random Walks in the plane with random rotations of past steps. By combining a complex martingale approach in the diffusive regime with a Poissonized branching-process framework in the spiraling regime, it establishes a complete central limit theorem: diffusive behavior for $\Re(\Phi_1)<1/2$, a $\sqrt{n\log n}$-scaling at the critical point $\Re(\Phi_1)=1/2$, and a randomly rotated logarithmic spiral when $\Re(\Phi_1)>1/2$, with an explicit limiting complex random variable $W$ and Gaussian fluctuations around $e^{\Phi_1\log n}W$ in the superdiffusive case. The analysis rests on a discrete complex martingale structure, detailed quadratic-variation estimates, and a branching Markov-process representation via Poissonization and the Yule process, yielding precise moment formulas for $W$ and a rate of convergence for the additive martingale. Altogether, the paper uncovers a phase transition in two dimensions and characterizes the spiral scaling regime through a rich interplay between reinforced random walks, complex-parameter branching, and urn-type dynamics.

Abstract

We consider in this article an Elephant Random Walk evolving in the plane. Specifically, this is a reinforced stochastic process in which the $n$th step is given by a random rotation of one of the previous steps chosen uniformly at random. We obtain a central limit theorem for this process, which shows that the process follows a randomly rotated logarithmic spiral at large times, with Gaussian fluctuations.

Figures (1)

• Figure 1: Sample path of $1000$ steps of an elephant random walks in $\mathbb{R}^2$ as defined in \ref{['eqn:defERWC']}, in which $\theta_1$ is a.s. a constant.

Theorems & Definitions (25)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof