Decorrelation of a leader by the increasing number of followers
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| Format: | Preprint |
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2024
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| author | Majumdar, Satya N. Schehr, Gregory |
| author_facet | Majumdar, Satya N. Schehr, Gregory |
| contents | 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 $τ= \sqrt{t_1/t_2}$ 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 maximises the exponent $γ$ and this maximal value freezes to $γ= 1/3$ for $τ>τ_c$. For the OU process, we show that $γ= {\rm tanh}(μ\,|t_1-t_2|/2)$ where $μ$ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_06964 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decorrelation of a leader by the increasing number of followers Majumdar, Satya N. Schehr, Gregory Statistical Mechanics Mathematical Physics Probability 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 $τ= \sqrt{t_1/t_2}$ 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 maximises the exponent $γ$ and this maximal value freezes to $γ= 1/3$ for $τ>τ_c$. For the OU process, we show that $γ= {\rm tanh}(μ\,|t_1-t_2|/2)$ where $μ$ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions. |
| title | Decorrelation of a leader by the increasing number of followers |
| topic | Statistical Mechanics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2403.06964 |