Can coarse-graining introduce long-range correlations in a symbolic sequence?

TL;DR

This paper develops an exactly solvable mean-field theory for correlated ternary sequences, revealing how coarse-graining can induce long-range correlations not present in the original system.

Contribution

It introduces a novel analytical framework for ternary sequences and explores how coarse-graining affects correlation properties, especially the mapping to binary sequences.

Findings

Variance shows linear or superlinear dependence on sequence length

Phase space includes diffusive and superdiffusive regions

Mapping between ternary and binary sequences reveals induced long-range correlations

Abstract

We present an exactly solvable mean-field-like theory of correlated ternary sequences which are actually systems with two independent parameters. Depending on the values of these parameters, the variance on the average number of any given symbol shows a linear or a superlinear dependence on the length of the sequence. We have shown that the available phase space of the system is made up a diffusive region surrounded by a superdiffusive region. Motivated by the fact that the diffusive portion of the phase space is larger than that for the binary, we have studied the mapping between these two. We have identified the region of the ternary phase space, particularly the diffusive part, that gets mapped into the superdiffusive regime of the binary. This exact mapping implies that long-range correlation found in a lower dimensional representative sequence may not, in general, correspond to the correlation properties of the original system.