Benjamini, Pemantle and Peres constructed nearest neighbor processes which have predictability profiles that decay faster than that of the simple random walk. Häggström and Mossel found processes with even faster decaying predictability profiles. We prove that the rate of decay achieved by Häggström and Mossel is optimal.
"Unpredictable nearest neighbor processes." Ann. Probab. 26 (4) 1781 - 1787, October 1998. https://doi.org/10.1214/aop/1022855882