Algebraic classification of actions invariant under generalized flip moves of two-dimensional graphs

Abstract

Statistical models defined on 2-dimensional graphs are classified which are invariant under flip moves, i.e., certain local changes of the adjacency structure of the graphs. The special case of regular graphs of degree 3—which are duals of 2-dimensional triangulations—corresponds to topological models and the classification leads to metrized, associative algebras. As a novel feature flip invariant models on regular graphs of degree 4 are classified by Z2-graded metrized associative algebras. They give rise to invariants for checkered graphs. Moreover, the general case of graphs with vertices of arbitrary degree (where degree 3 does occur) is discussed. Using structure theorems for (graded,) metrized, associative algebras we prove that only the simple ideals contribute to the partition function of such models. The partition functions are computed explicitly and reveal the invariant structures of the graph under the flip moves.