Paradoxical probabilistic behavior for strongly correlated many-body classical systems

TL;DR

This paper demonstrates that in strongly correlated classical systems, a small subset of variables can become asymptotically independent despite strong correlations, challenging traditional expectations.

Contribution

It introduces a probabilistic model showing paradoxical independence in strongly correlated systems and discusses implications for phase transition experiments.

Findings

Small fixed subset of variables becomes independent as system size grows.

Correlations persist when the subset size grows with system size.

Potential experimental verification near critical points.

Abstract

Using a simple probabilistic model, we illustrate that a small part of a strongly correlated many-body classical system can show a paradoxical behavior, namely asymptotic stochastic independence. We consider a triangular array such that each row is a list of $n$ strongly correlated random variables. The correlations are preserved even when $n\to\infty$, since the standard central limit theorem does not hold for this array. We show that, if we choose a fixed number $m<n$ of random variables of the $n$th row and trace over the other $n-m$ variables, and then consider $n\to\infty$, the $m$ chosen ones can, paradoxically, turn out to be independent. However, the scenario can be different if $m$ increases with $n$. Finally, we suggest a possible experimental verification of our results near criticality of a second-order phase transition.