Local-global correlations of dynamics on disordered energy landscapes

TL;DR

This paper investigates why the effective potential in Markov chains on energy landscapes correlates strongly with local exit rates, revealing conditions under which this correlation is high and providing bounds for specific models.

Contribution

It demonstrates that high correlation occurs when barrier heights vary less than well depths and extends the analysis to the Bouchaud trap model.

Findings

High correlation when barrier height variation is small compared to well depths

Bound on expected correlation for the Bouchaud trap model

Method combining lower bounds with Gaussian concentration inequality

Abstract

The stationary distribution of a continuous-time Markov chain generally arises from a complicated global balance of probability fluxes. Nevertheless, empirical evidence shows that the effective potential, defined as the negative logarithm of the stationary distribution, is often highly correlated with a simple local property of a state: the logarithm of its exit rate. To better understand why and how typically this correlation is high, we study reversible reaction kinetics on energy landscapes consisting of Gaussian wells and barriers, respectively associated with the vertices and edges of regular graphs. We find that for the correlation to be high it suffices for the heights of the barriers to vary significantly less than the depths of the wells, regardless of the degree of the underlying graph. As an application, we bound below the expected correlation exhibited by dynamics of the random energy model, known as the Bouchaud trap model. We anticipate that the proof, which combines a general lower bound of the expected correlation with the Gaussian concentration inequality, can be extended to several other classes of models.