J. Moles | Searcharxiv

arXiv2020

Gaussian concentration and uniqueness of equilibrium states in lattice systems

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space $S^{\mathbb{Z}^d}$ where $d\geq 1$ and $S$ is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.

math.PR↗

arXiv2019

Gaussian Concentration bound for potentials satisfying Walters condition with subexponential continuity rates

We consider the full shift $T:Ω\toΩ$ where $Ω=A^{\mathbb N}$, $A$ being a finite alphabet. For a class of potentials which contains in particular potentials $ϕ$ with variation decreasing like $O(n^{-α})$ for some $α>2$, we prove that their corresponding equilibrium state $μ_ϕ$ satisfies a Gaussian concentration bound. Namely, we prove that there exists a constant $C>0$ such that, for all $n$ and for all separately Lipschitz functions $K(x_0,\ldots,x_{n-1})$, the exponential moment of $K(x,\ldots,T^{n-1}x)-\int K(y,\ldots,T^{n-1}y)\, \mathrm{d}μ_ϕ(y)$ is bounded by $\exp\big(C\sum_{i=0}^{n-1}\mathrm{Lip}_i(K)^2\big)$. The crucial point is that $C$ is independent of $n$ and $K$. We then derive various consequences of this inequality. For instance, we obtain bounds on the fluctuations of the empirical frequency of blocks, the speed of convergence of the empirical measure, and speed of Markov approximation of $μ_ϕ$. We also derive an almost-sure central limit theorem.

math.DS↗