Decorrelation of a leader by an increasing number of followers

We compute the connected two-time correlator of the maximum MN(t) of N independent Gaussian stochastic processes (GSPs) characterized 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 process (OUP). 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 maximizes the exponent γ and this maximal value freezes to γ=1/3 for τ>τc. For the OUP, we show that γ=tanh(μ0.16em0ex|t₁-t₂|/2), where μ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions.