Uniform LSI for the canonical ensemble on the 1d-lattice with strong, finite-range interaction

TL;DR

This paper proves a uniform logarithmic Sobolev inequality for a one-dimensional lattice system with unbounded, real-valued spins and strong finite-range interactions, extending classical and recent results.

Contribution

It extends the uniform LSI to unbounded, real-valued spins with strong interactions, using a novel combination of two-scale and block-decomposition methods.

Findings

LSI constant is uniform in boundary data and external field

LSI scales optimally with system size

Extension of classical results to unbounded spins with strong interactions

Abstract

We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show that the canonical ensemble (ce) satisfies a uniform logarithmic Sobolev inequality (LSI). The LSI constant is uniform in the boundary data, the external field and scales optimally in the system size. This extends a classical result of H.T. Yau from discrete to unbounded, real-valued spins. It also extends prior results of Landim, Panizo & Yau or Menz for unbounded, real-valued spins from absent- or weak- to strong-interaction. The proof of the LSI uses a combination of the two-scale approach and a block-decomposition technique introduced by Zegarlinski. Main ingredients are the strict convexity of the coarse-grained Hamiltonian, the equivalence of ensembles and the decay of correlations in the ce. Those ingredients were recently provided by the authors.