## Brazilian Journal of Probability and Statistics

### Unions of random walk and percolation on infinite graphs

Kazuki Okamura

#### Abstract

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.

#### Article information

Source
Braz. J. Probab. Stat., Volume 33, Number 3 (2019), 586-637.

Dates
Accepted: May 2018
First available in Project Euclid: 10 June 2019

https://projecteuclid.org/euclid.bjps/1560153854

Digital Object Identifier
doi:10.1214/18-BJPS404

Mathematical Reviews number (MathSciNet)
MR3960278

Zentralblatt MATH identifier
07094819

Keywords
Bernoulli percolation random walk

#### Citation

Okamura, Kazuki. Unions of random walk and percolation on infinite graphs. Braz. J. Probab. Stat. 33 (2019), no. 3, 586--637. doi:10.1214/18-BJPS404. https://projecteuclid.org/euclid.bjps/1560153854

#### References

• 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 Physics 130, 983–1009.
• Asselah, A. and Schapira, B. (2017a). Boundary of the range of transient random walk. Probability Theory and Related Fields 168, 691–719.
• Asselah, A. and Schapira, B. (2017b). Moderate deviations for the range of a transient random walk: Path concentration. Annales Scientifiques de l’Ecole Normale Supérieure 50, 755–786.
• 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 Physics 278, 385–431.
• Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge: Cambridge University Press.
• Brézis, H. and Lieb, E. (1983). A relation between pointwise convergence of functions and convergence of functionals. Proceedings of the American Mathematical Society 88, 486–490.
• Diestel, R. (2010). Graph Theory, 4th ed. Heidelberg: Springer.
• 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 Mathematics 32, 721–747.
• 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.
• Fitzner, R. and van der Hofstad, R. (2017). Mean-field behavior for nearest-neighbor percolation in $d>10$. Electronic Journal of Probability 22, 1–65.
• Fontes, L. and Newman, C. M. (1983). First passage percolation for random colorings of $\mathbb{Z}^{d}$. The Annals of Applied Probability 3, 746–762. Erratum: The Annals of Applied Probability 4 (1994), 254.
• Gibson, L. R. (2008). The mass of sites visited by a random walk on an infinite graph. Electronic Communications in Probability 13, 1257–1282.
• Grimmett, G. R. (1999). Percolation, 2nd ed. Berlin: Springer.
• Grimmett, G. R. and Piza, M. S. T. (1997). Decay of correlations in random-cluster model. Communications in Mathematical Physics 189, 465–480.
• Hamana, Y. (2001). Asymptotics of the moment generating function for the range of random walks. Journal of Theoretical Probability 14, 189–197.
• Heydenreich, M., van der Hofstad, R. and Hulshof, T. (2014). Random walk on the high-dimensional IIC. Communications in Mathematical Physics 329, 57–115.
• Higuchi, Y. and Wu, X.-Y. (2008). Uniqueness of the critical probability for percolation in the two-dimensional Sierpiński carpet lattice. Kobe Journal of Mathematics 25, 1–24.
• Jain, N. C. and Pruitt, W. E. (1970). The range of recurrent random walk in the plane. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 16, 279–292.
• 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.
• Kesten, H. and Spitzer, F. (1963). Ratio theorems for random walks I. Journal d’Analyse Mathématique 11, 285–321.
• Kozma, G. and Nachmias, A. (2009). The Alexander–Orbach conjecture holds in high dimensions. Inventiones Mathematicae 178, 635–654.
• Kumagai, T. (1997). Percolation on pre-Sierpinski carpets. In New Trends in Stochastic Analysis (Charingworth, 1994), 288–304. River Edge, NJ: World Sci. Publ.
• Kumagai, T. (2014). Random Walks on Disordered Media and Their Scaling Limits. Lecture Notes in Mathematics 2101. Cham: Springer.
• Kumagai, T. and Misumi, J. (2008). Heat kernel estimates for strongly recurrent random walk on random media. Journal of Theoretical Probability 21, 910–935.
• Lawler, G. (1996). Intersections of Random Walks. New York: Birkhäuser.
• Le Gall, J.-F. (1986a). Proprietés d’intersection des marches aléatoires. Communications in Mathematical Physics 104, 471–507.
• Le Gall, J.-F. (1986b). Sur la saucisse de Wiener et les points multiples du mouvement brownien. The Annals of Probability 14, 1219–1244.
• Le Gall, J.-F. (1988). Fluctuation results for the Wiener sausage. The Annals of Probability 16, 991–1018.
• Liggett, T. M. (1985). An improved subadditive ergodic theorem. The Annals of Probability 13, 1279–1285.
• Okada, I. (2016). The inner boundary of random walk range. Journal of the Mathematical Society of Japan 68, 939–959.
• Okamura, K. (2014). On the range of random walk on graphs satisfying a uniform condition. ALEA. Latin American Journal of Probability and Mathematical Statistics 11, 341–357.
• Okamura, K. (2017). Enlargement of subgraphs of infinite graphs by Bernoulli percolation. Indagationes Mathematicae (New Series) 28, 832–853.
• Okamura, K. (2018). Long time behavior of the volume of the Wiener sausage on Dirichlet spaces. Potential Analysis. To appear.
• Port, S. C. (1965). Limit theorems involving capacities for recurrent Markov chains. Journal of Mathematical Analysis and Applications 12, 555–569.
• Port, S. C. (1966). Limit theorems involving capacities. Journal of Mathematics and Mechanics 15, 805–832.
• Port, S. C. and Stone, C. J. (1968). Hitting times for transient random walks. Journal of Mathematics and Mechanics 17, 1117–1130.
• Port, S. C. and Stone, C. J. (1969). Potential theory of random walks on Abelian groups. Acta Mathematica 122, 19–114.
• Shinoda, M. (1996). Percolation on the pre-Sierpinski gasket. Osaka Journal of Mathematics 33, 533–554.
• Shinoda, M. (2002). Existence of phase transition of percolation on Sierpinski carpet lattices. Journal of Applied Probability 39, 1–10.
• Shinoda, M. (2003). Non-existence of phase transition of oriented percolation on Sierpinski carpet lattices. Probability Theory and Related Fields 125, 447–456.
• Spitzer, F. (1964). Electrostatic capacity, heat flow and Brownian motion. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 3, 110–121.
• Spitzer, F. (1976). Principles of Random Walk, 2nd ed. New York: Springer.
• Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge: Cambridge University Press.