Self-avoiding walks and polygons on hyperbolic graphs

We prove that for the d $d$-regular tessellations of the hyperbolic plane by k $k$-gons, there are exponentially more self-avoiding walks of length n $n$ than there are self-avoiding polygons of length n $n$. 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 $k$, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion d - 1 - O ( 1 / d ) $d-1-O(1\unicode{x02215}d)$ as d -> infinity $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 $n$ is comparable to the n $n$th power of their connective constant. Some of these results were previously obtained by Madras and Wu for all but finitely many regular tessellations of the hyperbolic plane.