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From Maximum of Inter-Visit Times to Starving Random Walks
Léo Régnier1, Maxim Dolgushev1, Olivier Bénichou1
1Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne University, 4 Place Jussieu, 75005 Paris, France.
Physical Review Letters
|April 5, 2024
Summary
We analyzed the longest duration for a random walk to find new sites, crucial for understanding exploration. This research solves the starving random walk problem across various diffusion types.
Area of Science:
- Physics
- Mathematics
- Statistical Mechanics
Background:
- A new observable, the time to visit a novel site (τk), quantifies random walk exploration.
- The study focuses on the statistics of Mn, the maximum of these novel site visit times (τ0 to τn-1).
Purpose of the Study:
- To determine the asymptotic statistics of Mn, the longest duration between novel site visits.
- To apply these findings to foraging theory and solve the d-dimensional starving random walk problem.
Main Methods:
- Analysis of extreme value statistics for correlated and non-identically distributed random variables.
- Asymptotic determination of the statistics of Mn.
Main Results:
- The statistical properties of Mn were asymptotically determined.
- Explicit solutions were found for the d-dimensional starving random walk problem.
Conclusions:
- The study provides a fundamental understanding of random walk exploration and extreme value statistics.
- The methods and results have direct applications in foraging and diverse diffusion processes.
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