Decorrelation of a leader by the increasing number of followers

Abstract

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 $\rho$ 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 $N^{-\gamma }$ (with logarithmic corrections) with a decorrelation exponent $\gamma =(1-\rho )/(1+\rho )$ that depends only on $\rho$, 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, $\rho$ is only a function of $\tau =\sqrt{t_1/t_2}$ and we find an interesting ``freezing'' transition at a critical value $\tau ={\tau }_c=(3-\sqrt{5})/2$. For $\tau <{\tau }_c$, there is an optimal $H^{\ast }(\tau )>0$ that maximises the exponent $\gamma$ and this maximal value freezes to $\gamma =1/3$ for $\tau >{\tau }_c$. For the OU process, we show that $\gamma =\tanh (\mu |t_1-t_2|/2)$ where $\mu$ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions.

arXiv commentMain text: 6 pages, 3 figures. Supplementary material: 10 pages

References in corpus (28)

• Forecast and Control of Epidemics in a Globalized World
• Intermittent search strategies
• Stochastic Resetting and Applications
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Experimental realization of diffusion with stochastic resetting
• Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results
• Cleaning large correlation matrices: tools from random matrix theory
• Persistence exponents for fluctuating interfaces
• Extreme value statistics of correlated random variables: a pedagogical review
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Generalized Elastic Model yields Fractional Langevin Equation
• Statistics of Lead Changes in Popularity-Driven Systems
• Record Statistics for Multiple Random Walks
• Evolutionary trajectories in rugged fitness landscapes
• Punctuated evolution for the quasispecies model
• Universal Extremal Statistics in a Freely Expanding Jepsen Gas
• Temporal correlations of the running maximum of a Brownian trajectory
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• First-Passage Exponents of Multiple Random Walks
• Effusion of stochastic processes on a line
• Probabilistic Phase Space Trajectory Description for Anomalous Polymer Dynamics
• Anomalous Fluctuations of Extremes in Many-Particle Diffusion
• Slow Kinetics of Brownian Maxima
• Maximal distance travelled by N vicious walkers till their survival
• On the Mixing of Diffusing Particles
• Maxima of Two Random Walks: Universal Statistics of Lead Changes
• Scaling Exponents for Ordered Maxima
• Infinite System of Random Walkers: Winners and Losers

Cited by in corpus (1)

• Universality of Top Rank Statistics for Brownian Reshuffling