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\times N) square with free boundary, where at each step the walker chooses uniformly among all legal king moves. \frac{8}{3\pi}N^2\log^2 N \frac{4}{3\pi}N^2\log N\log\log N+o!\left(N^2\log N\log\log N\right).] 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.