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

TL;DR

This paper introduces a method to analytically solve a broad class of strongly correlated systems by integrating over a random parameter, demonstrated through physical examples like particles undergoing various motions with resetting.

Contribution

The paper presents a novel analytical framework for strongly correlated systems by leveraging conditional independence and random parameters, enabling exact solutions for complex observables.

Findings

Exact solutions for sum and extreme statistics in correlated systems

Verification through numerical simulations

Applicable to diverse physical models with resetting mechanisms

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 $\{X_1,\, X_2,\ldots, X_N\}$ 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) L\'evy 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.