This preprint was accepted January, 1998.
Contact: A.M.Vershik
Abstract:
We provide a full large deviation principle (LDP)
for the uniform measure on certain ensembles of convex
lattice polygons.
This LDP provides for the analysis of concentration of
the measure on convex closed curves. In particular,
convergence
to a limiting shape results in some particular cases, including
convergence to a circle when the ensemble is defined
as those centered convex polygons with total length bounded by a constant.
The Gauss-Minkowskii transform of convex curves plays a
crucial role in our approach.