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Stationary coalescing walks on the lattice

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Abstract

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.

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Notes

  1. See Lions and Souganidis [15] in the context of continuum stochastic homogenization.

References

  1. Ahlberg, D., Hoffman, C.: Random coalescing geodesics in first-passage percolation. ArXiv e-prints (2016)

  2. Baxter, J.R.: A class of ergodic automorphisms. ProQuest LLC, Ann Arbor. Ph.D. thesis, University of Toronto, Canada (1969)

  3. Benjamini, I., Tessera, R.: First Passage percolation on a hyperbolic graph admits bi-infinite geodesics (2016). arXiv:1606.02449 [math]

  4. Burton, R.M., Keane, M.: Density and uniqueness in percolation. Commun. Math. Phys. 121(3), 501–505 (1989)

    Article  MathSciNet  Google Scholar 

  5. Chacon, R.V.: Weakly mixing transformations which are not strongly mixing. Proc. Am. Math. Soc. 22, 559–562 (1969)

    Article  MathSciNet  Google Scholar 

  6. Damron, M., Hanson, J.: Busemann functions and infinite geodesics in two-dimensional first-passage percolation. Commun. Math. Phys. 325(3), 917–963 (2014)

    Article  MathSciNet  Google Scholar 

  7. Garet, O., Marchand, R.: Coexistence in two-type first-passage percolation models. Ann. Appl. Probab. 15(1A), 298–330 (2005)

    Article  MathSciNet  Google Scholar 

  8. Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Geodesics and the competition interface for the corner growth model (2015). arXiv:1510.00860 [math]

  9. Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Variational formulas and cocycle solutions for directed polymer and percolation models. Commun. Math. Phys. 346(2), 741–779 (2016)

    Article  MathSciNet  Google Scholar 

  10. Hoffman, C.: Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15(1B), 739–747 (2005)

    Article  MathSciNet  Google Scholar 

  11. Kesten, H.: Aspects of first passage percolation. In: Hennequin, P.L. (ed.) École d’été de probabilités de Saint-Flour, XIV-1984, Volume 1180 of Lecture Notes in Mathematics, pp. 125–264. Springer, Berlin (1986)

    Google Scholar 

  12. Kingman, J.F.C.: The ergodic theory of subadditive stochastic processes. J. R. Stat. Soc. Ser. B. Methodol. 30, 499–510 (1968)

    MathSciNet  MATH  Google Scholar 

  13. Krishnan, A.: Variational formula for the time constant of first-passage percolation. Commun. Pure. Appl. Math. 69(10), 1984–2012 (2016)

    Article  MathSciNet  Google Scholar 

  14. Licea, C., Newman, C.M.: Geodesics in two-dimensional first-passage percolation. Ann. Probab. 24(1), 399–410 (1996)

    Article  MathSciNet  Google Scholar 

  15. Lions, P.-L., Souganidis, P.E.: Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting. Commun. Pure Appl. Math. 56(10), 1501–1524 (2003)

    Article  MathSciNet  Google Scholar 

  16. Newman, C.M.: A surface view of first-passage percolation. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994), Basel, pp. 1017–1023. Birkhäuser (1995)

  17. Newman, C.M., Schulman, L.S.: Infinite clusters in percolation models. J. Stat. Phys. 26(3), 613–628 (1981)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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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  1. University of Utah, Salt Lake City, UT, USA

    Jon Chaika

  2. University of Rochester, Rochester, NY, USA

    Arjun Krishnan

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  1. Jon Chaika
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  2. Arjun Krishnan
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Correspondence to Arjun Krishnan.

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Chaika, J., Krishnan, A. Stationary coalescing walks on the lattice. Probab. Theory Relat. Fields 175, 655–675 (2019). https://doi.org/10.1007/s00440-018-0893-2

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