## The Annals of Probability

### Painting a graph with competing random walks

Jason Miller

#### Abstract

Let $X_{1}$, $X_{2}$ be independent random walks on $\mathbf{Z} _{n}^{d}$, $d\geq3$, each starting from the uniform distribution. Initially, each site of $\mathbf{Z} _{n}^{d}$ is unmarked, and, whenever $X_{i}$ visits such a site, it is set irreversibly to $i$. The mean of $|\mathcal{A} _{i}|$, the cardinality of the set $\mathcal{A} _{i}$ of sites painted by $i$, once all of $\mathbf{Z} _{n}^{d}$ has been visited, is $\frac{1}{2}n^{d}$ by symmetry. We prove the following conjecture due to Pemantle and Peres: for each $d\geq3$ there exists a constant $\alpha_{d}$ such that $\lim_{n\to\infty}\operatorname{Var} (|\mathcal{A} _{i}|)/h_{d}(n)=\frac{1}{4}\alpha_{d}$ where $h_{3}(n)=n^{4}$, $h_{4}(n)=n^{4}(\log n)$ and $h_{d}(n)=n^{d}$ for $d\geq5$. We will also identify $\alpha_{d}$ explicitly and show that $\alpha_{d}\to1$ as $d\to\infty$. This is a special case of a more general theorem which gives the asymptotics of $\operatorname{Var} (|\mathcal{A} _{i}|)$ for a large class of transient, vertex transitive graphs; other examples include the hypercube and the Caley graph of the symmetric group generated by transpositions.

#### Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 636-670.

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.aop/1362750937

Digital Object Identifier
doi:10.1214/11-AOP713

Mathematical Reviews number (MathSciNet)
MR3077521

Zentralblatt MATH identifier
1271.05091

#### Citation

Miller, Jason. Painting a graph with competing random walks. Ann. Probab. 41 (2013), no. 2, 636--670. doi:10.1214/11-AOP713. https://projecteuclid.org/euclid.aop/1362750937

#### References

• [1] Bhattacharya, R. N. and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York.
• [2] Brummelhuis, M. J. A. M. and Hilhorst, H. J. (1991). Covering of a finite lattice by a random walk. Phys. A 176 387–408.
• [3] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2001). Thick points for planar Brownian motion and the Erdős–Taylor conjecture on random walk. Acta Math. 186 239–270.
• [4] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2004). Cover times for Brownian motion and random walks in two dimensions. Ann. of Math. (2) 160 433–464.
• [5] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2006). Late points for random walks in two dimensions. Ann. Probab. 34 219–263.
• [6] Dembo, A. and Sznitman, A.-S. (2006). On the disconnection of a discrete cylinder by a random walk. Probab. Theory Related Fields 136 321–340.
• [7] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.
• [8] Dicker, L. (2006). Coloring a $d\geq3$ dimensional lattice with two independent random walks. M.S. thesis, Univ. Pennsylvania. Available at http://www.math.upenn.edu/~pemantle/papers/Student-theses/Masters/Dicker060421.pdf.
• [9] Gomes, S. R. Jr., Lucena, L. S., da Silva, L. R. and Hilhorst, H. J. (1996). Coloring of a one-dimensional lattice by two independent random walkers. Phys. A 225 81–88.
• [10] Jain, N. and Orey, S. (1968). On the range of random walk. Israel J. Math. 6 373–380.
• [11] Jain, N. C. and Pruitt, W. E. (1970). The central limit theorem for the range of transient random walk. Bull. Amer. Math. Soc. (N.S.) 76 758–759.
• [12] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
• [13] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
• [14] Matthews, P. (1988). Covering problems for Markov chains. Ann. Probab. 16 1215–1228.
• [15] Miller, J. and Peres, Y. (2012). Uniformity of the uncovered set of random walk and cutoff for lamplighter chains. Ann. Probab. 40 535–577.
• [16] Pemantle, R. and Peres, Y. Private communication.
• [17] Sznitman, A.-S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 2039–2087.
• [18] Windisch, D. (2008). Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 140–150.
• [19] Windisch, D. (2010). Random walks on discrete cylinders with large bases and random interlacements. Ann. Probab. 38 841–895.