 94406 Cesi F, Martinelli F
 On the Layering Transition of an SOS
Surface Interacting with a Wall. I. Equilibrium Results
(194K, TeX)
Dec 23, 94

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Abstract. We consider the model of a 2D surface above a fixed
wall and attracted towards it by means of a positive magnetic field
$h$ in the solid on solid (SOS) approximation, when the inverse
temperature $\beta$ is very large
and the external
field $h$ is exponentially small in $\b$.
We improve considerably previous results
by Dinaburg and Mazel on the competition between the
external field and the entropic repulsion with the wall, leading,
in this case, to the phenomenon of layering phase transitions.
In particular we show, using the Pirogov Sinai scheme as
given by Zahradn\'\i k, that there exists a
unique critical value $h^*_k(\beta)$ in the
interval $({1\over 4}e^{4\beta k}, 4e^{4\beta k})$
such that, for all $h\in (h^*_{k+1},h^*_k)$ and $\beta$ large
enough, there exists a unique infinite volume Gibbs state.
The typical configurations are small perturbations of
the ground state represented by a surface at height $k+1$ above the
wall. Moreover, for the same choice of the thermodynamic parameters,
the influence of the boundary conditions of the Gibbs measure
in a finite cube decays exponentially fast with the
distance from the boundary.
When $h=h^*_k(\beta)$ we prove instead the convergence of the
cluster expansion for both $k$ and $k+1$ boundary conditions. This
fact signals the presence of a phase transition.
In the second paper
of this series we will consider a Glauber dynamics for the above model
and we will study the rate of approach to equilibrium in a
large finite cube
with arbitrary boundary conditions as a function of
the external field $h$.
Using
the results proven in this paper we will show that there is a dramatic
slowing down in the approach to equilibrium when the magnetic field takes one
of the critical values and the boundary conditions are free (absent).
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