Decorrelation of a leader by an increasing number of followers.

Abstract

We compute the connected two-time correlator of the maximum M_{N}(t) of N independent Gaussian stochastic processes (GSPs) characterized by a common correlation coefficient ρ 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^{-γ} (with logarithmic corrections) with a decorrelation exponent γ=(1-ρ)/(1+ρ) that depends only on ρ, 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 process (OUP). For the fBm, ρ is only a function of τ=sqrt[t_{1}/t_{2}] and we find an interesting freezing transition at a critical value τ=τ_{c}=(3-sqrt[5])/2. For τ<τ_{c}, there is an optimal H^{*}(τ)>0 that maximizes the exponent γ and this maximal value freezes to γ=1/3 for τ>τ_{c}. For the OUP, we show that γ=tanh(μ|t_{1}-t_{2}|/2), where μ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions.