Abstract:
How long does a self-avoiding walk on a finite transitive > graph have to be before it `feels’ the effect of the finite volume? We > discuss the elementary exact solution for self-avoiding walk on the > complete graph, which shows a separation into a dilute phase, a critical > window, and a dense phase. The exact solution on the complete graph > provides a prototype for self-avoiding walk on other high-dimensional > finite transitive graphs, including the hypercube and discrete tori in > dimensions 
, and we discuss recent results for the dilute phase > obtained in these two settings via the lace expansion. Related work in > progress with Jiwoon Park (University of Cambridge) on the finite-size > scaling of the 
-dimensional hierarchical 
-component 
> spin model will also be mentioned.
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Based on
https://arxiv.org/abs/2008.00080
https://arxiv.org/abs/2203.07695