Self‐avoiding walks and polygons on hyperbolic graphs

Abstract

AbstractWe prove that for the ‐regular tessellations of the hyperbolic plane by ‐gons, there are exponentially more self‐avoiding walks of length than there are self‐avoiding polygons of length . 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 , we show that the connective constant for self‐avoiding walks satisfies the asymptotic expansion as ; 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 is comparable to the 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.