QUANTITATIVE SUB-BALLISTICITY OF SELF-AVOIDING WALK ON THE HEXAGONAL LATTICE

We prove quantitative sub-ballisticity for the self-avoiding walk on the hexagonal lattice. Namely, we show that, with high probability, a self-avoiding walk of length n does not exit a ball of radius O(n/logn). Previously, only a nonquantitative o(n) bound was known from the work of Duminil-Copin and Hammond (Comm. Math. Phys. 324 (2013) 401–423). As an important ingredient of the proof, we show that at criticality the partition function of bridges of height T decays polynomially fast to 0 as T tends to infinity, which we believe to be of independent interest.