47 research outputs found
On the zero-temperature limit of Gibbs states
We exhibit Lipschitz (and hence H\"older) potentials on the full shift
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 for which the
zero-temperature limit of the associated Gibbs measures does not exist. In
higher dimension, namely on the configuration space ,
, we show that this non-convergence behavior can occur for finite-range
interactions, that is, for locally constant potentials.Comment: The statement of Theorem 1.2 is more accurate and some new comment
follow i
Realization of aperiodic subshifts and uniform densities in groups
A theorem of Gao, Jackson and Seward, originally conjectured to be false by
Glasner and Uspenskij, asserts that every countable group admits a
-coloring. A direct consequence of this result is that every countable group
has a strongly aperiodic subshift on the alphabet . In this article,
we use Lov\'asz local lemma to first give a new simple proof of said theorem,
and second to prove the existence of a -effectively closed strongly
aperiodic subshift for any finitely generated group . We also study the
problem of constructing subshifts which generalize a property of Sturmian
sequences to finitely generated groups. More precisely, a subshift over the
alphabet has uniform density if for every
configuration the density of 's in any increasing sequence of balls
converges to . We show a slightly more general result which implies
that these subshifts always exist in the case of groups of subexponential
growth.Comment: minor typos correcte