Mean-field behavior for the survival probability and the percolation point-to-surface connectivity

We consider the critical survival probability (up to timet) for oriented percolation and the contact process, and the point-to-surface (of the ball of radiust) connectivity for critical percolation. Let t denote both quantities. We prove in a unified fashion that, if t exhibits a power law and both the two-point function and its certain restricted version exhibit the same mean-field behavior, then t=t-1 for the time-oriented models with d > 4 and t=t-2 for percolation with d > 7.