On Constrained Annealed Bounds for Pinning and Wetting Models
Francesco Caravenna, Università di Milano-Bicocca, Italy
Giambattista Giacomin, Universitè Paris 7 and Laboratoire de Probabilités et Modèles Aléatoires C.N.R.S., France
Abstract
The free energy of quenched disordered systems is bounded above by
the free energy of the corresponding annealed system. This bound may
be improved by applying the annealing procedure, which is just
Jensen inequality, after having modified the Hamiltonian in a way
that the quenched expressions are left unchanged. This procedure is
often viewed as a partial annealing or as a constrained annealing,
in the sense that the term that is added may be interpreted as a
Lagrange multiplier on the disorder variables.
In this note we point out that, for a family of models, some of
which have attracted much attention, the multipliers of the form of
empirical averages of local functions cannot improve on the basic
annealed bound from the viewpoint of characterizing the phase
diagram. This class of multipliers is the one that is suitable for
computations and it is often believed that in this class one can
approximate arbitrarily well the quenched free energy.
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