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

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.

Equivalence of the grand canonical ensemble and the canonical ensemble on 1d-lattice systems

“…Lemma 2.4 (Lemma 3.1 in [KM18b]). There is a constant C ∈ (0, ∞) which is uniform in the system size N and the external fields s, σ such that…”

“…While the decay of correlation of an ensemble itself is a very interesting property, it also plays an integral role in deducing a uniform log-Sobolev inequality (LSI) of the ensemble. Indeed, in the case of strong, finite-range interactions, the uniform LSI of the ce was deduced in [KM18b]. Moreover, it was also shown with the help of decay of correlation that the ce on the one-dimensional lattice does not have a phase transition (see [KM19]).…”

Decay of Correlations and Uniqueness of the Infinite-Volume Gibbs Measure of the Canonical Ensemble of 1d-Lattice Systems

“…We expect that this is a manifestation of a more general principle for systems on general lattices: If the gce has sufficient decay of correlations then the equivalence of correlations of the gce and ce should hold (see also [CT04]). In [KM18b], the decay of correlations of the ce (cf. Theorem 2.9) is an important ingredient in the proof that the ce with arbitrary strong, ferromagnetic, finite-range interaction satisfies a uniform log-Sobolev inequality on the one-dimensional lattice.…”