Distribution of the time between maximum and minimum of random walks

We consider a one-dimensional Brownian motion of fixed duration T. Using a path-integral technique, we compute exactly the probability distribution of the difference tau = t(min)-t(max) between the time t(min )of the global minimum and the time t(max) of the global maximum. We extend this result to a Brownian bridge, i.e., a periodic Brownian motion of period T. In both cases, we compute analytically the first few moments of tau, as well as the covariance of t(max) and t(min) , showing that these times are anticorrelated. We demonstrate that the distribution of tau for Brownian motion is valid for discrete-time random walks with n steps and with a finite jump variance, in the limit n -> infinity. In the case of Levy flights, which have a divergent jump variance, we numerically verify that the distribution of tau differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "tau = n" is exactly 1/(2n) for any finite n, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of (1 + 1)-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size L. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys.Rev.Lett.123,200201 (2019)].