Universal distribution of the number of minima for random walks and Lévy flights
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View abstract on PubMed
Summary
This summary is machine-generated.The number of local minima in random landscapes is fully characterized for random walks and Lévy flights. Strikingly, this distribution is universal, independent of jump specifics, and differs between fixed-step and first-passage ensembles.
Area Of Science
- Statistical Physics
- Complex Systems
- Stochastic Processes
Background
- Understanding the statistical properties of random processes is crucial in various scientific fields.
- One-dimensional landscapes generated by random walks and Lévy flights are fundamental models for studying complex systems.
- The number of local minima in such landscapes is a key characteristic that influences system behavior.
Purpose Of The Study
- To compute the exact full distribution of the number of local minima (m) in one-dimensional landscapes.
- To investigate these distributions for two distinct ensembles: fixed number of steps (N) and first-passage time to the origin.
- To determine the universality and dependence on jump distribution for these minima distributions.
Main Methods
- Analytical computation of the distribution of local minima.
- Consideration of two distinct ensembles: fixed-N and first-passage time.
- Comparison of results for random walks and Lévy flights.
Main Results
- The distribution of local minima (m) differs significantly between the fixed-N ensemble (Gaussian) and the first-passage time ensemble (power-law tail m^{-3/2}).
- A key finding is the universality of these distributions for all m, irrespective of the specific jump distribution.
- The distributions for Lévy flights and random walks with finite jump variance are identical.
Conclusions
- The number of local minima in random one-dimensional landscapes exhibits universal statistical properties.
- The choice of ensemble (fixed-N vs. first-passage time) dictates the nature of the minima distribution.
- These universal properties hold true for both random walks and Lévy flights, simplifying the understanding of complex landscape statistics.
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