Self-avoiding walk models on mean-field graphs

We study various walk models on the complete graph and the full m-ary tree. In particular, we study the mean and variance asymptotics of these models, as well as the weak limits. We study these properties for a sequence of fugacities; a parameter used when studying variable-length ensembles to represent the distribution weight. While some results on finite graphs are already established, we observed a critical window for fugacities that converge quickly enough to the critical point, and a scaling window which smoothly links the critical and non-critical regions. These critical and scaling windows are established for the mean and variance asymptotics. We also show universality between weakly self-avoiding walks and self-avoiding walks on the complete graph.