Decorrelation of a leader by the increasing number of followers

We compute the connected two-time correlator of the maximum MN(t) of N independent Gaussian stochastic processes (GSP) characterised by a common correlation coefficient ρ that depends on the two times t₁ and t₂. We show analytically that this correlator, for fixed times t₁ and t₂, 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 = √t₁/t₂ and we find an interesting ``freezing'' transition at a critical value τ= τc=(3-√5)/2. For τ < τc, there is an optimal H^*(τ) > 0 that maximises the exponent γ and this maximal value freezes to γ= 1/3 for τ >τc. For the OU process, we show that γ = tanh(μ \,|t₁-t₂|/2) where μ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions.