Exponential growth rate of lattice comb polymers

We investigate a lattice model of comb polymers and derive bounds on the exponential growth rate of the number of embeddings of the comb. A comb is composed of a backbone that is a self-avoiding walk and a set of t teeth, also modelled as mutually and self-avoiding walks, attached to the backbone at vertices or nodes of degree 3. Each tooth of the comb has m a edges and there are m b edges in the backbone between adjacent degree 3 vertices and between the first and last nodes of degree 3 and the end vertices of degree 1 of the backbone. We are interested in the exponential growth rate as t → ∞ with m a and m b fixed. We prove upper bounds on this growth rate and show that for small values of m a the growth rate is strictly less than that of self-avoiding walks.