Exponential growth rate of lattice comb polymers

EJ Janse van Rensburg; SG Whittington

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arXiv · 2407.21112

Exponential growth rate of lattice comb polymers

EJ Janse van Rensburg·SG Whittington

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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.

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EJ Janse van Rensburg, SG Whittington. 2024-07-30. Exponential growth rate of lattice comb polymers. https://arxiv.org/abs/2407.21112

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