Exact extreme, order and sum statistics in a class of strongly correlated system

Abstract

Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations.. We consider a set of $N$ independent and identically distributed (i.i.d) random variables $\{X_1,X_2,\dots ,X_N\}$ whose common distribution has a parameter $Y$ (or a set of parameters) which itself is random with its own distribution. For a fixed value of this parameter $Y$, the $X_i$ variables are independent and we call them conditionally independent and identically distributed (c.i.i.d). However, once integrated over the distribution of the parameter $Y$, the $X_i$ variables get strongly correlated, yet retaining a solvable structure for various observables, such as for the sum and the extremes of $X_i$'s. This provides a simple procedure to generate a class of solvable strongly correlated systems. We illustrate how this procedure works via three physical examples where $N$ particles on a line perform independent (i) Brownian motions, (ii) ballistic motions with random initial velocities, and (iii) Lévy flights, but they get strongly correlated via {\it simultaneous resetting} to the origin. Our results are verified in numerical simulations. This procedure can be used to generate an endless variety of solvable strongly correlated systems.

arXiv comment26 pages, 9 figures

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• Dynamically generated correlations in a trapped bosonic gas via frequency quenches
• Diffusion with stochastic resetting on a lattice
• The distribution of the maximum of independent resetting Brownian motions
• Experimental evidence for strong emergent correlations between particles in a switching trap