Decorrelation of a leader by the increasing number of followers

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 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 (OU) process. For the fBm, ρ is only a function of τ = √(t_1/t_2) and we find an interesting “freezing” transition at a critical value τ= τ_c=(3-√(5))/2. For τ < τ_c, there is an optimal H^*(τ) > 0 that maximises the exponent γ and this maximal value freezes to γ= 1/3 for τ >τ_c. For the OU process, 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.