A sharp estimate for cover times on binary trees

We compute the second order correction for the cover time of the binary tree of depth n by (continuous-time) random walk, and show that with probability approaching 1 as n increases, τ cov = | E | [ 2 log 2 ⋅ n − log n / 2 log 2 + O ( ( log log n ) 8 ) ] , thus showing that the second order correction differs from the corresponding one for the maximum of the Gaussian free field on the tree.