On the zero-temperature limit of Gibbs states
arXiv:0907.0081 · doi:10.1007/s00220-010-0997-8
Abstract
We exhibit Lipschitz (and hence Hölder) potentials on the full shift $\{0,1\}^{\mathbb{N}}$ such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space $\{0,1\}^{\mathbb Z}$ for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space $\{0,1\}^{\mathbb{Z}^{d}}$, $d\geq3$, we show that this non-convergence behavior can occur for finite-range interactions, that is, for locally constant potentials.
The statement of Theorem 1.2 is more accurate and some new comment follow it