Two Combinatorial Models with identical Statics yet different Dynamics

TL;DR

This paper introduces two combinatorial models with identical static properties but different dynamic behaviors, illustrating how local minima affect relaxation times in sorting-related systems.

Contribution

The paper presents two models with the same spectra but differing dynamics, highlighting the impact of local minima on relaxation processes.

Findings

One model relaxes slowly to sorted state

The other model gets blocked by local minima

Models share identical spectra and partition functions

Abstract

Motivated by the problem of sorting, we introduce two simple combinatorial models with distinct Hamiltonians yet identical spectra (and hence partition function) and show that the local dynamics of these models are very different. After a deep quench, one model slowly relaxes to the sorted state whereas the other model becomes blocked by the presence of stable local minima.