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 $H^{\ast }(\tau )>0$ 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 $\gamma =\tanh (\mu |t_1-t_2|/2)$ where $\mu$ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions.

Main text: 6 pages, 3 figures. Supplementary material: 10 pages

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