Exponential growth rate of lattice comb polymers

TL;DR

This paper studies the exponential growth rate of lattice comb polymers, deriving bounds and comparing it to self-avoiding walks, revealing that for small teeth sizes the growth rate is lower.

Contribution

It provides new bounds on the growth rate of comb polymers and compares it to self-avoiding walks, advancing understanding of their combinatorial complexity.

Findings

Upper bounds on the exponential growth rate are established.

For small tooth sizes, the growth rate is strictly less than that of self-avoiding walks.

The growth rate depends on the number of teeth and their size.

Abstract

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 $t \to \infty$ 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.