Tony Guttmann (University of Melbourne)
In 1962 Hammersely and Welsh proved that for self-avoiding walks in dimension d≥2, there exists a positive constant C such that the number of SAWs cn of length n satisfies $c_n \le \exp{C\sqrt{n}} \mu^n$ for all n∈ℕ. There has been little improvement in the ensuing 60 years. Recently Duminil-Copin, Ganguly, Hammond and Manolescu made a small improvement in which \sqrt{n} is replaced by $\sqrt{n-\epsilon}$. I will describe three approaches (largely due to Hammersely in private discussions with me) to improving this bound to a power law, so that the bound would then be $c_n \le C\mu^n n^g$. Indeed, it is universally believed that, for d=2, that is the asymptotic form, with g=11/32.