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We give Harnack inequalities for the hitting distributions of a large familyof symmetric random walks on $\Z^2$, and their projections onto the latticetorus $\Z^2_K$. This extends a framework for the simple random walk in Dembo,et al. (2006), and generalizes the results in Bass, Rosen (2007) to the toralprojection.

Harnack Inequalities of Hitting Distributions of Projections of Planar Symmetric Random Walks on the Lattice Torus