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Unions of random walk and percolation on infinite graphs. (English) Zbl 1427.60078
Summary: We consider a random object that is associated with both random walks and random media, specifically, the superposition of a configuration of subcritical Bernoulli percolation on an infinite connected graph and the trace of the simple random walk on the same graph. We investigate asymptotics for the number of vertices of the enlargement of the trace of the walk until a fixed time, when the time tends to infinity. This process is more highly self-interacting than the range of random walk, which yields difficulties. We show a law of large numbers on vertex-transitive transient graphs. We compare the process on a vertex-transitive graph with the process on a finitely modified graph of the original vertex-transitive graph and show their behaviors are similar. We show that the process fluctuates almost surely on a certain non-vertex-transitive graph. On the two-dimensional integer lattice, by investigating the size of the boundary of the trace, we give an estimate for variances of the process implying a law of large numbers. We give an example of a graph with unbounded degrees on which the process behaves in a singular manner. As by-products, some results for the range and the boundary, which will be of independent interest, are obtained.
MSC:
 60G50 Sums of independent random variables; random walks 60K35 Interacting random processes; statistical mechanics type models; percolation theory 05C81 Random walks on graphs
Keywords:
Bernoulli percolation; random walk
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References:
 [1] Antunović, T. and Veselić, I. (2008). Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs. Journal of Statistical Physics130, 983-1009. [2] Asselah, A. and Schapira, B. (2017a). Boundary of the range of transient random walk. Probability Theory and Related Fields168, 691-719. · Zbl 1382.60066 [3] Asselah, A. and Schapira, B. (2017b). Moderate deviations for the range of a transient random walk: Path concentration. Annales Scientifiques de l’Ecole Normale Supérieure50, 755-786. · Zbl 1378.60055 [4] Barlow, M. T., Járai, A. A., Kumagai, T. and Slade, G. (2008). Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Communications in Mathematical Physics278, 385-431. · Zbl 1144.82030 [5] Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge: Cambridge University Press. [6] Brézis, H. and Lieb, E. (1983). A relation between pointwise convergence of functions and convergence of functionals. Proceedings of the American Mathematical Society88, 486-490. · Zbl 0526.46037 [7] Diestel, R. (2010). Graph Theory, 4th ed. Heidelberg: Springer. · Zbl 1204.05001 [8] Donsker, M. D. and Varadhan, S. R. S. (1979). On the number of distinct sites visited by a random walk. Communications on Pure and Applied Mathematics32, 721-747. · Zbl 0418.60074 [9] Dvoretzky, A. and Erdös, P. (1951). Some problems on random walk in space. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 353-368. Berkeley, CA: Univ. California Press. [10] Fitzner, R. and van der Hofstad, R. (2017). Mean-field behavior for nearest-neighbor percolation in $$d>10$$. Electronic Journal of Probability22, 1-65. · Zbl 1364.60130 [11] Fontes, L. and Newman, C. M. (1983). First passage percolation for random colorings of $$\mathbb{Z}^{d}$$. The Annals of Applied Probability3, 746-762. Erratum: The Annals of Applied Probability4 (1994), 254. · Zbl 0780.60101 [12] Gibson, L. R. (2008). The mass of sites visited by a random walk on an infinite graph. Electronic Communications in Probability13, 1257-1282. · Zbl 1191.60059 [13] Grimmett, G. R. (1999). Percolation, 2nd ed. Berlin: Springer. · Zbl 0926.60004 [14] Grimmett, G. R. and Piza, M. S. T. (1997). Decay of correlations in random-cluster model. Communications in Mathematical Physics189, 465-480. · Zbl 0888.60084 [15] Hamana, Y. (2001). Asymptotics of the moment generating function for the range of random walks. Journal of Theoretical Probability14, 189-197. · Zbl 0979.60034 [16] Heydenreich, M., van der Hofstad, R. and Hulshof, T. (2014). Random walk on the high-dimensional IIC. Communications in Mathematical Physics329, 57-115. · Zbl 1302.82050 [17] Higuchi, Y. and Wu, X.-Y. (2008). Uniqueness of the critical probability for percolation in the two-dimensional Sierpiński carpet lattice. Kobe Journal of Mathematics25, 1-24. · Zbl 1169.82006 [18] Jain, N. C. and Pruitt, W. E. (1970). The range of recurrent random walk in the plane. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete16, 279-292. · Zbl 0194.49205 [19] Jain, N. C. and Pruitt, W. E. (1972). The range of random walk. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III, 31-50. Berkeley, CA: Univ. California Press. [20] Kesten, H. and Spitzer, F. (1963). Ratio theorems for random walks I. Journal d’Analyse Mathématique11, 285-321. · Zbl 0121.35201 [21] Kozma, G. and Nachmias, A. (2009). The Alexander-Orbach conjecture holds in high dimensions. Inventiones Mathematicae178, 635-654. · Zbl 1180.82094 [22] Kumagai, T. (1997). Percolation on pre-Sierpinski carpets. In New Trends in Stochastic Analysis (Charingworth, 1994), 288-304. River Edge, NJ: World Sci. Publ. [23] Kumagai, T. (2014). Random Walks on Disordered Media and Their Scaling Limits. Lecture Notes in Mathematics2101. Cham: Springer. · Zbl 1360.60003 [24] Kumagai, T. and Misumi, J. (2008). Heat kernel estimates for strongly recurrent random walk on random media. Journal of Theoretical Probability21, 910-935. · Zbl 1159.60029 [25] Lawler, G. (1996). Intersections of Random Walks. New York: Birkhäuser. · Zbl 0925.60078 [26] Le Gall, J.-F. (1986a). Proprietés d’intersection des marches aléatoires. Communications in Mathematical Physics104, 471-507. · Zbl 0609.60078 [27] Le Gall, J.-F. (1986b). Sur la saucisse de Wiener et les points multiples du mouvement brownien. The Annals of Probability14, 1219-1244. · Zbl 0621.60083 [28] Le Gall, J.-F. (1988). Fluctuation results for the Wiener sausage. The Annals of Probability16, 991-1018. · Zbl 0665.60080 [29] Liggett, T. M. (1985). An improved subadditive ergodic theorem. The Annals of Probability13, 1279-1285. · Zbl 0579.60023 [30] Okada, I. (2016). The inner boundary of random walk range. Journal of the Mathematical Society of Japan68, 939-959. · Zbl 1359.60061 [31] Okamura, K. (2014). On the range of random walk on graphs satisfying a uniform condition. ALEA. Latin American Journal of Probability and Mathematical Statistics11, 341-357. · Zbl 1296.05179 [32] Okamura, K. (2017). Enlargement of subgraphs of infinite graphs by Bernoulli percolation. Indagationes Mathematicae (New Series)28, 832-853. · Zbl 1367.05147 [33] Okamura, K. (2018). Long time behavior of the volume of the Wiener sausage on Dirichlet spaces. Potential Analysis. To appear. [34] Port, S. C. (1965). Limit theorems involving capacities for recurrent Markov chains. Journal of Mathematical Analysis and Applications12, 555-569. · Zbl 0134.34905 [35] Port, S. C. (1966). Limit theorems involving capacities. Journal of Mathematics and Mechanics15, 805-832. · Zbl 0146.38406 [36] Port, S. C. and Stone, C. J. (1968). Hitting times for transient random walks. Journal of Mathematics and Mechanics17, 1117-1130. · Zbl 0162.49201 [37] Port, S. C. and Stone, C. J. (1969). Potential theory of random walks on Abelian groups. Acta Mathematica122, 19-114. · Zbl 0183.47201 [38] Shinoda, M. (1996). Percolation on the pre-Sierpinski gasket. Osaka Journal of Mathematics33, 533-554. · Zbl 0870.60094 [39] Shinoda, M. (2002). Existence of phase transition of percolation on Sierpinski carpet lattices. Journal of Applied Probability39, 1-10. · Zbl 1009.60086 [40] Shinoda, M. (2003). Non-existence of phase transition of oriented percolation on Sierpinski carpet lattices. Probability Theory and Related Fields125, 447-456. · Zbl 1020.60098 [41] Spitzer, F. (1964). Electrostatic capacity, heat flow and Brownian motion. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete3, 110-121. · Zbl 0126.33505 [42] Spitzer, F. (1976). Principles of Random Walk, 2nd ed. New York: Springer. · Zbl 0359.60003 [43] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge: Cambridge University Press. · Zbl 0951.60002
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