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 mₐ edges and there are mb 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 mₐ and mb fixed. We prove upper bounds on this growth rate and show that for small values of mₐ the growth rate is strictly less than that of self-avoiding walks.