Decorrelation of a leader by the increasing number of followers

Abstract

We compute the connected two-time correlator of the maximum $M_N(t)$ of $N$ independent Gaussian stochastic processes (GSP) characterised by a common correlation coefficient $\rho$ that depends on the two times $t_1$ and $t_2$. We show analytically that this correlator, for fixed times $t_1$ and $t_2$, decays for large $N$ as a power law $N^{-\gamma}$ (with logarithmic corrections) with a decorrelation exponent $\gamma = (1-\rho)/(1+ \rho)$ that depends only on $\rho$, but otherwise is universal for any GSP. We study several examples of physical processes including the fractional Brownian motion (fBm) with Hurst exponent $H$ and the Ornstein-Uhlenbeck (OU) process. For the fBm, $\rho$ is only a function of $\tau = \sqrt{t_1/t_2}$ and we find an interesting ``freezing'' transition at a critical value $\tau= \tau_c=(3-\sqrt{5})/2$. For $\tau < \tau_c$, there is an optimal $H^*(\tau) > 0$ that maximises the exponent $\gamma$ and this maximal value freezes to $\gamma= 1/3$ for $\tau >\tau_c$. For the OU process, we show that $\gamma = {\rm tanh}(\mu \,|t_1-t_2|/2)$ where $\mu$ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions.