Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures

Yannic Steenbeck

Abstract

We prove that there is only one translation-invariant Gibbsian point process w.r.t. to a chosen interaction if any of them satisfies a certain bound related to concentration-of-measure. This concentration-of-measure bound is e.g. fulfilled if a corresponding modified logarithmic Sobolev inequality holds. In particular, for natural examples with non-uniqueness regimes, a modified logarithmic Sobolev inequality cannot be satisfied. Therefore, in these situations, the free-energy dissipation in related continuous-time birth-and-death dynamics in $\mathbb{R}^d$ is not exponentially fast.

Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures

Abstract

We prove that there is only one translation-invariant Gibbsian point process w.r.t. to a chosen interaction if any of them satisfies a certain bound related to concentration-of-measure. This concentration-of-measure bound is e.g. fulfilled if a corresponding modified logarithmic Sobolev inequality holds. In particular, for natural examples with non-uniqueness regimes, a modified logarithmic Sobolev inequality cannot be satisfied. Therefore, in these situations, the free-energy dissipation in related continuous-time birth-and-death dynamics in is not exponentially fast.

Paper Structure

This paper contains 4 sections, 7 theorems, 67 equations.

Table of Contents

  1. Modified logarithmic Sobolev inequalities, related concentration bounds and uniqueness of Gibbs measures via specific relative entropy distance
  2. Proofs
  3. Dirichlet form with birth rate $1$
  4. Dirichlet form with birth rate $b$

Key Result

Theorem 1.4

[theorem]theorem:MLSI_implies_distance_in_specific_relative_entropy Let $\nu \in \mathcal{P}_\theta$ satisfy ( MLSI--1) with some constant $c_\nu > 0$. Then, $\mathscr{I}(\mu \,\vert\, \nu) > 0$ for all $\mu \in \mathcal{P}_\theta \setminus \{\nu\}$.

Theorems & Definitions (15)

  • Example 1.1: Very nice pair potentials
  • Example 1.2: Area interaction
  • Example 1.3: Superstable pair interactions
  • Theorem 1.4: MLSI--1 implies strictly positive specific relative entropy distance
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7: MLSI--b implies strictly positive specific relative entropy distance
  • Remark 1.8
  • Proposition 1.9: MGF bounds imply distance in specific relative entropy
  • Proposition 1.10: MGF bounds from MLSI--1
  • ...and 5 more