# Self-avoiding walks and polygons on hyperbolic graphs

## TL;DR

This paper investigates the properties of self-avoiding walks and polygons on hyperbolic graphs, establishing exponential growth differences, ballistic behavior, and asymptotic properties of connective constants, extending prior results to a broader class of tessellations.

## Contribution

It proves exponential growth of walks over polygons on hyperbolic tessellations, shows walks are ballistic, and analyzes asymptotic behavior of connective constants, generalizing previous work.

## Key findings

- Self-avoiding walks grow exponentially faster than polygons on hyperbolic graphs.
- Self-avoiding walks exhibit ballistic behavior on vertex-transitive graphs.
- Connective constants for walks and polygons have distinct asymptotic behaviors.

## Abstract

We prove that for the $d$-regular tessellations of the hyperbolic plane by $k$-gons, there are exponentially more self-avoiding walks of length $n$ than there are self-avoiding polygons of length $n$. We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed $k$, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion $d-1-O(1/d)$ as $d\to \infty$; on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self-avoiding walks of length $n$ is comparable to the $n$th power of their connective constant. Some of these results were previously obtained by Madras and Wu \cite{MaWuSAW} for all but finitely many regular tessellations of the hyperbolic plane.