The Random King's Cover Time on a Square: Second-order asymptotics and exact free-boundary constants

We study the cover time of a random king walk on an (N× N) square with free boundary, where at each step the walker chooses uniformly among all legal king moves. 8/3^2log^2 N 4/3^2log Nloglog N+o!(N^2log Nloglog N).] The proof combines potential-kernel estimates for the finite-range king walk, multiscale traversal processes, critical branching-process barrier estimates, two-point correlation analysis, excursion-time control, and a treatment of the free boundary, including sides and corners. We also derive the exact Green-function structure at a flat free boundary. In particular, the logarithmic Green coefficient at the boundary is exactly twice its bulk value. Numerical simulations are included to illustrate finite-size behavior and the size of the remaining third-order correction. The work provides a concrete finite-range, free-boundary extension of two-dimensional cover-time methods and highlights the strong influence of boundary geometry on late points and finite-size corrections.