Supermultiplicative relations in models of interacting self-avoiding walks and polygons

TL;DR

This paper generalizes Fekete's lemma using supermultiplicative relations to prove the existence of thermodynamic limits in models of interacting self-avoiding walks and polygons, supported by numerical estimations.

Contribution

It introduces a generalized supermultiplicative relation that ensures the existence of limits in models of self-avoiding walks and polygons, extending classical results.

Findings

Existence of thermodynamic limits for models of adsorbing walks and polygons.

Numerical estimates of the shape of the limit function for lattice models.

Generalization of Fekete's lemma to two-variable supermultiplicative relations.

Abstract

Fekete's lemma shows the existence of limits in subadditive sequences. This lemma, and generalisations of it, also have been used to prove the existence of thermodynamic limits in statistical mechanics. In this paper it is shown that the two variable supermultiplicative relation $$ p_{n_1}(m_1)\,p_{n_2}(m_2) \leq p_{n_1+n_2}(m_1 + m_2) $$ together with mild assumptions, imply the existence of the limit $$ \log \mathcal{P}_\#(\epsilon) = \lim_{n\to\infty} \frac{1}{n} \log p_n(\lfloor \epsilon n \rfloor).$$ This is a generalisation of Fekete's lemma. The existence of $\mathcal{P}_\#(\epsilon)$ are proven for models of adsorbing walks and polygons, and for pulled polygons. In addition, numerical data are presented estimating the general shape of $\log \mathcal{P}_\# (\epsilon)$ of models of square lattice self-avoiding walks and polygons.