Abstract
We present an exact solution for the probability density function of the time difference between the minimum and the maximum of a one-dimensional Brownian motion of duration . We then generalize our results to a Brownian bridge, i.e., a periodic Brownian motion of period . We demonstrate that these results can be directly applied to study the position difference between the minimal and the maximal heights of a fluctuating ()-dimensional Kardar-Parisi-Zhang interface on a substrate of size , in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for Lévy flights and find that it differs from the Brownian motion result.
- Received 13 September 2019
DOI:https://doi.org/10.1103/PhysRevLett.123.200201
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