Measures on contour, polymer or animal models. A probabilistic approach

Abstract

We present a new approach to study measures on ensembles of contours, polymers or other objects interacting by some sort of exclusion condition. For concreteness we develop it here for the case of Peierls contours. Unlike existing methods, which are based on cluster-expansion formalisms and/or complex analysis, our method is strictly probabilistic and hence can be applied even in the absence of analyticity properties. It involves a Harris graphical construction of a loss network for which the measure of interest is invariant. The existence of the process and its mixing properties depend on the absence of infinite clusters for a dual (backwards) oriented percolation process which we dominate by a multitype branching process. Within the region of subcriticality of this branching process the approach yields: (i) exponential convergence to the equilibrium (=contour) measures, (ii) standard clustering and finite-effect properties of the contour measure, (iii) a particularly strong form of the central limit theorem, and (iv) a Poisson approximation for the distribution of contours at low temperature.