Convergence to equilibrium for a directed (1+d)-dimensional polymer

TL;DR

This paper analyzes the mixing time of a directed (1+d)-dimensional lattice path model, showing it scales like L^2 log L, and establishes a logarithmic Sobolev inequality across dimensions.

Contribution

It extends the understanding of mixing times and inequalities from 1D to higher dimensions for lattice path models, using induction and transposition estimates.

Findings

Mixing time scales as L^2 log L with a d-dependent constant.

Logarithmic Sobolev inequality holds on diffusive scale L^2 for fixed d.

Lower bounds are established via Wilson's argument.

Abstract

We consider a flip dynamics for directed (1+d)-dimensional lattice paths with length L. The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter corresponding to the case d=1. We prove that the mixing time of the associated Markov chain scales like L^2\log L up to a d-dependent multiplicative constant. The key step in the proof of the upper bound is to show that the system satisfies a logarithmic Sobolev inequality on the diffusive scale L^2 for every fixed d, which we achieve by a suitable induction over the dimension together with an estimate for adjacent transpositions. The lower bound is obtained with a version of Wilson's argument for the one-dimensional case.