A corner transfer matrix renormalisation group investigation of the vertex-interacting self-avoiding walk model

A recently introduced extension of the Corner Transfer Matrix Renormalisation Group (CTMRG) method useful for the study of self-avoiding walk type models is presented in detail and applied to a class of interacting self-avoiding walks due to Bl\"ote and Nienhuis. This model displays two different types of collapse transition depending on model parameters. One is the standard $\theta$-point transition. The other is found to give rise to a first-order collapse transition despite being known to be in other respects critical.