Abstract
We consider spin-flip dynamics of Ising lattice spin systems and study the time evolution of concentration inequalities. For “weakly interacting” dynamics we show that the Gaussian concentration bound is conserved in the course of time and it is satisfied by the unique stationary Gibbs measure. Next we show that, for a general class of translation-invariant spin-flip dynamics, it is impossible to evolve in finite time from a low-temperature Gibbs state towards a measure satisfying the Gaussian concentration bound. Finally, we consider the time evolution of the weaker uniform variance bound, and show that this bound is conserved under a general class of spin-flip dynamics.
Highlights
Concentration inequalities are important tools to understand the fluctuation properties of general observables f (σ1, . . . , σn) which are functions of n random variables (σ1, . . . , σn), where n is large but finite
More precisely we study the following questions in the context of spin-flip dynamics of lattice spin systems: 1. When started from a probability measure satisfying the Gaussian concentration bound, do we have this bound at later times?
Guided by the intuition coming from this context, one expects that a high-temperature dynamics should conserve the Gaussian concentration bound. We prove this result in the present paper, using the expansion in [10], i.e., under the condition that the flip rates are sufficiently close to the rates of an independent spin-flip dynamics
Summary
Concentration inequalities are important tools to understand the fluctuation properties of general observables f (σ1, . . . , σn) which are functions of n random variables (σ1, . . . , σn), where n is large but finite. Concentration inequalities are important tools to understand the fluctuation properties of general observables f Σn) which are functions of n random variables Σn), where n is large but finite. For bounded random variables which are independent (or weakly dependent) typically one can obtain so-called Gaussian concentration bounds for the fluctuations of f In the context of lattice spin systems, one has, e.g., σi ∈ {−1, +1}, with i ∈ [−n, n]d ∩ Zd , and these random variables are distributed according to a Gibbs measure. The “weak dependence” between them means for instance that we are in the Dobrushin uniqueness regime, which is for instance the case at “high enough”.
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