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Quantitative Russo–Seymour–Welsh for Random Walk on Random Graphs and Decorrelation of Uniform Spanning Trees

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Abstract

We prove a quantitative Russo–Seymour–Welsh (RSW)-type result for random walks on two natural examples of random planar graphs: the supercritical percolation cluster in \(\mathbb {Z}^2\) and the Poisson Voronoi triangulation in \(\mathbb {R}^2\). More precisely, we prove that the probability that a simple random walk crosses a rectangle in the hard direction with uniformly positive probability is stretched exponentially likely in the size of the rectangle. As an application, we prove a near optimal decorrelation result for uniform spanning trees for such graphs. This is the key missing step in the application of the proof strategy of Berestycki et al. (Ann Probab 48(1):1–52, 2020) for such graphs [in Berestycki et al. (2020), random walk RSW was assumed to hold with probability 1]. Applications to almost sure Gaussian-free field scaling limit for dimers on Temperleyan-type modification on such graphs are also discussed.

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Notes

  1. x, y are \(*\)-connected by closed vertices, if there is a path of closed vertices with two consecutive vertices at distance either 1 or 2 in \(\mathbb {Z}^2\) (i.e., diagonally adjacent or regular adjacent) connecting x and y. It is well-known that a regular cluster is blocked by a \(*\)-connected circuit.

  2. This part of the bound is probably not optimal, but since we will be content with a stretched exponential bound anyway in the end, we do not pursue to make this optimal.

  3. Euclidean diameter of a set \(A \subset \mathbb {R}^2\) is \(\sup \{|x-y|:x,y \in A\}\).

References

  1. Antal, P., Pisztora, A.: On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24(2), 1036–1048 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barlow, M.T.: Random walks on supercritical percolation clusters. Ann. Probab. 32(4), 3024–3084 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berestycki, N., Laslier, B., Ray, G.: A note on dimers and T-graphs. arXiv:1610.07994 (2016)

  4. Berestycki, N., Laslier, B., Ray, G.: Dimers and imaginary geometry. Ann. Probab. 48(1), 1–52 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berestycki, N., Laslier, B., Ray, G.: The dimer model on Riemann surfaces, I: Temperleyan forests. arXiv:1908.00832 (2021)

  6. Berestycki, N., Laslier, B., Ray, G.: The dimer model on Riemann surfaces II: scaling limit. (2022). arXiv:2207.09875

  7. Berger, N., Biskup, M.: Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Relat. Fields 137(1–2), 83–120 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chelkak, D., Laslier, B., Russkikh, M.: Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces. arXiv:2109.06272

  9. Chelkak, D., Laslier, B., Russkikh, M.: Dimer model and holomorphic functions on t-embeddings of planar graphs. arXiv:2001.11871

  10. Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228(3), 1590–1630 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. de Tilière, B.: The dimer model in statistical mechanics. Lecture notes, available on the author’s webpage

  12. Duminil-Copin, H.: Lectures on the Ising and Potts models on the hypercubic lattice. In: PIMS-CRM Summer School in Probability, pp. 35–161. Springer, Berlin (2017)

  13. Duminil-Copin, H.: Sixty years of percolation. In: Proceeding of the International Congress. World Scientific (2018)

  14. Garet, O., Marchand, R.: Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 35(3), 833–866 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grimmett, G.: What is percolation? In: Percolation, pp. 1–31. Springer, Berlin (1999)

  16. Kenyon, R.: Dominos and the Gaussian free field. Ann. Probab. 29(3), 1128–1137 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kenyon, R.W., Propp, J.G., Wilson, D.B.: Trees and matchings. Electron. J. Combin. 7 (2000)

  18. Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab. 25(1), 71–95 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lyons, R., Peres, Y.: Probability on Trees and Networks, vol. 42. Cambridge University Press, Cambridge (2017)

    MATH  Google Scholar 

  20. Rousselle, A.: Quenched invariance principle for random walks on Delaunay triangulations. Electronic J. Probab. 20, 1–32 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 296–303 (1996)

  22. Yadin, A., Yehudayoff, A.: Loop-erased random walk and Poisson kernel on planar graphs. Ann. Probab. 39(4), 1243–1285 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Benoit Laslier for several useful discussions.

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  1. University of Victoria, Victoria, Canada

    Gourab Ray & Tingzhou Yu

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  1. Gourab Ray
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  2. Tingzhou Yu
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Correspondence to Gourab Ray.

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Gourab Ray: Supported in part by NSERC 50311-57400.

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Ray, G., Yu, T. Quantitative Russo–Seymour–Welsh for Random Walk on Random Graphs and Decorrelation of Uniform Spanning Trees. J Theor Probab 36, 2284–2310 (2023). https://doi.org/10.1007/s10959-023-01248-7

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