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How Long does it Take to Train an Elephant Random Walk.

¹Institute of mathematics, University of Zurich, Winterthurerstrasse 190, Zürich, 8057 Zürich Switzerland.

Journal of Statistical Physics

|June 22, 2026

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This study investigates how initial training impacts the Elephant Random Walk

Area of Science:

Stochastic processes
Mathematical physics

Background:

The Elephant Random Walk (ERW) is a model of animal foraging behavior.
Understanding the influence of initial conditions on random walks is crucial for modeling complex systems.

Purpose of the Study:

To analyze the effect of conditioning on the first k steps (training) on the long-term behavior of the ERW.
To quantify the elephant's 'forgetting rate' based on its return time to the origin.

Main Methods:

Mathematical analysis of the Elephant Random Walk model.
Derivation of scaling laws for the first return time to the origin under specific initial conditions.

Main Results:

The first return time to the origin scales as k^((4-4p)/(3-4p)) in the diffusive regime.

Keywords:

First return time Martingale Scaling limit Training

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The first return time grows exponentially in the critical regime.

These results provide a measure of how quickly the elephant forgets its initial training.

Conclusions:

The initial training significantly influences the long-term dynamics of the Elephant Random Walk.
The rate of forgetting is dependent on the specific regime (diffusive or critical).