Tricritical points of trails, their Euler digraphs and graphs: Exact results on the Sierpinski gasket

The collapse transitions of trails, their oriented graphs and silhouettes (Eulerian digraphs and graphs, respectively), as the fugacity for crossings is increased, are investigated by exact decimation on the 2D Sierpinski gasket. Recursion relations between the generating functions for the three basic configurations on consecutive levels are derived. For all models the authors find tricritical points which move along a line in a four-dimensional parameter space, as the fugacity is varied, and terminate at a decoupled first- or second-order multicritical point. It suggests these models belong to distinct universality classes which differ from that of self-attracting polymer chains which do not undergo a collapse Theta transition on this fractal lattice.