Stationary coalescing walks on the lattice
We consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice. We assume that once walks meet, they coalesce. In 2d, we classify the collective behavior of these walks under mild assumptions: they either coalesce almost surely or form bi-infinite trajectories. Bi-infinite trajectories form measure-preserving dynamical systems, have a common asymptotic direction in 2d, and possess other nice properties. We use our theory to classify the behavior of compatible families of semi-infinite geodesics in stationary first- and last-passage percolation. We also partially answer a question raised by C. Hoffman about the limiting empirical measure of weights seen by geodesics. We construct several examples: our main example is a standard first-passage percolation model where geodesics coalesce almost surely, but have no asymptotic direction or average weight.
KeywordsMeasure preserving transformations Bi-infinite trajectories Geodesics First-passage percolation
Mathematics Subject Classification37A05 37A50 60K35 60K37
The authors thank Firas Rassoul-Agha, Timo Seppäläinen, Eric Cator, and Michael Damron for helpful conversations. J. Chaika was supported in part by NSF Grants DMS-135500 and DMS-1452762, the Sloan foundation and a Warnock chair. A. Krishnan was supported in part by an AMS Simons travel grant.
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