Exponential decay of pairwise correlation in Gaussian graphical models with an equicorrelational one-dimensional connection pattern

TL;DR

This paper demonstrates that in Gaussian graphical models with a one-dimensional equicorrelational structure, pairwise correlations decay exponentially with distance, providing insights into the behavior of large-scale models.

Contribution

It establishes the exponential decay of correlations in such models and analyzes the limit as the number of variables approaches infinity.

Findings

Correlation decays exponentially with distance

Derived the limit for infinite variables

Quantified the difference between finite and infinite cases

Abstract

We consider Gaussian graphical models associated with an equicorrelational and one-dimensional conditional independence graph. We show that pairwise correlation decays exponentially as a function of distance. We also provide a limit when the number of variables tend to infinity and quantify the difference between the finite and infinite cases.