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Facilitated oriented spin models:some non equilibrium results | | Nicoletta Cancrini
; Fabio Martinelli
; Roberto H. Schonmann
; Cristina Toninelli
; | | Date: |
23 Oct 2008 | | Abstract: | We analyze the relaxation to equilibrium for kinetically constrained spin
models (KCSM) when the initial distribution $
u$ is different from the
reversible one, $mu$. This setting has been intensively studied in the physics
literature to analyze the slow dynamics which follows a sudden quench from the
liquid to the glass phase. We concentrate on two basic oriented KCSM: the East
model on $bZ$, for which the constraint requires that the East neighbor of
the to-be-update vertex is vacant and the model on the binary tree introduced
in cite{Aldous:2002p1074}, for which the constraint requires the two children
to be vacant. While the former model is ergodic at any $p
eq 1$, the latter
displays an ergodicity breaking transition at $p_c=1/2$. For the East we prove
exponential convergence to equilibrium with rate depending on the spectral gap
if $
u$ is concentrated on any configuration which does not contain a forever
blocked site or if $
u$ is a Bernoulli($p’$) product measure for any $p’
eq
1$. For the model on the binary tree we prove similar results in the regime
$p,p’<p_c$ and under the (plausible) assumption that the spectral gap is
positive for $p<p_c$. By constructing a proper test function we also prove that
if $p’>p_c$ and $pleq p_c$ convergence to equilibrium cannot occur for all
local functions. Finally we present a very simple argument (different from the
one in cite{Aldous:2002p1074}) based on a combination of combinatorial results
and ’’energy barrier’’ considerations, which yields the sharp upper bound for
the spectral gap of East when $puparrow 1$. | | Source: | arXiv, 0810.4237 | | Services: | Forum | Review | PDF | Favorites | |
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