Relaxation time for a dimer covering with height representation

TL;DR

This study investigates the relaxation dynamics of dimer coverings on a square lattice, revealing that the slowest relaxation modes scale linearly with system size, with implications for understanding interface fluctuations.

Contribution

It provides an exact lower bound on relaxation time for the discrete dimer model, connecting continuum theory predictions with discrete system behavior.

Findings

Longest relaxation time scales as system size N

Two slow modes identified: height fluctuations and mean height drift

Exact lower bound of O(N) on relaxation time established

Abstract

This paper considers the Monte Carlo dynamics of random dimer coverings of the square lattice, which can be mapped to a rough interface model. Two kinds of slow modes are identified, associated respectively with long-wavelength fluctuations of the interface height, and with slow drift (in time) of the system-wide mean height. Within a continuum theory, the longest relaxation time for either kind of mode scales as the system size N. For the real, discrete model, an exact lower bound of O(N) is placed on the relaxation time, using variational eigenfunctions corresponding to the two kinds of continuum modes.