A combinatorial approach to branched polymers’ statistics

Abstract

At least ideally, a certain class of polymers presents itself as a collection (set) of connected components. Each of these components is a cycle of trees, that is branched polymers eventually rooted on a cycle. We derive (and study) an equilibrium statistical model that accounts for the main connectivity features of such structures, whose origin is to be found in combinatorial probability. Phase transition (gel–soltransition) is shown to occur when some internal control parameter crosses one (critical parameter). Various structural asymptotic results are shown to be available using singularity analysis.