## The Annals of Applied Probability

### Trapping games on random boards

#### Abstract

We consider the following two-player game on a graph. A token is located at a vertex, and the players take turns to move it along an edge to a vertex that has not been visited before. A player who cannot move loses. We analyze outcomes with optimal play on percolation clusters of Euclidean lattices.

On $\mathbb{Z}^{2}$ with two different percolation parameters for odd and even sites, we prove that the game has no draws provided closed sites of one parity are sufficiently rare compared with those of the other parity (thus favoring one player). We prove this also for certain $d$-dimensional lattices with $d\geq3$. It is an open question whether draws can occur when the two parameters are equal.

On a finite ball of $\mathbb{Z}^{2}$, with only odd sites closed but with the external boundary consisting of even sites, we identify up to logarithmic factors a critical window for the trade-off between the size of the ball and the percolation parameter. Outside this window, one or the other player has a decisive advantage.

Our analysis of the game is intimately tied to the effect of boundary conditions on maximum-cardinality matchings.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 6 (2016), 3727-3753.

Dates
Revised: February 2016
First available in Project Euclid: 15 December 2016

https://projecteuclid.org/euclid.aoap/1481792598

Digital Object Identifier
doi:10.1214/16-AAP1190

Mathematical Reviews number (MathSciNet)
MR3582816

Zentralblatt MATH identifier
1356.05088

#### Citation

Basu, Riddhipratim; Holroyd, Alexander E.; Martin, James B.; Wästlund, Johan. Trapping games on random boards. Ann. Appl. Probab. 26 (2016), no. 6, 3727--3753. doi:10.1214/16-AAP1190. https://projecteuclid.org/euclid.aoap/1481792598

#### References

• [1] Aizenman, M. and Grimmett, G. (1991). Strict monotonicity for critical points in percolation and ferromagnetic models. J. Stat. Phys. 63 817–835.
• [2] Anderson, W. N. Jr. (1974). Maximum matching and the game of Slither. J. Combin. Theory Ser. B 17 234–239.
• [3] Bondy, J. A. and Murty, U. S. R. (1976). Graph Theory with Applications. American Elsevier Publishing Co., New York.
• [4] Cerf, R. and Cirillo, E. N. M. (1999). Finite size scaling in three-dimensional bootstrap percolation. Ann. Probab. 27 1837–1850.
• [5] Cerf, R. and Manzo, F. (2002). The threshold regime of finite volume bootstrap percolation. Stochastic Process. Appl. 101 69–82.
• [6] Cornell Math Explorers’ Club. Available at http://www.math.cornell.edu/~mec/2003-2004/graphtheory/slither/howtoplayslither.html.
• [7] Fraenkel, A. S., Scheinerman, E. R. and Ullman, D. (1993). Undirected edge geography. Theoret. Comput. Sci. 112 371–381.
• [8] Froböse, K. (1989). Finite-size effects in a cellular automaton for diffusion. J. Stat. Phys. 55 1285–1292.
• [9] Gardner, M. (1972). Mathematical games. Sci. Am. 227 176–182.
• [10] Holroyd, A. E. (2003). Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields 125 195–224.
• [11] Holroyd, A. E. (2006). The metastability threshold for modified bootstrap percolation in $d$ dimensions. Electron. J. Probab. 11 418–433 (electronic).
• [12] Holroyd, A. E., Marcovici, I. and Martin, J. B. (2015). Percolation games, probabilistic cellular automata, and the hardcore model. Preprint. Available at arXiv:1503.05614.
• [13] Holroyd, A. E. and Martin, J. B. Galton–Watson games. In preparation.
• [14] Hopcroft, J. E. and Karp, R. M. (1973). An $n^{5/2}$ algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2 225–231.
• [15] Lichtenstein, D. and Sipser, M. (1980). GO is polynomial-space hard. J. ACM 27 393–401.
• [16] Renault, G. and Schmidt, S. (2015). On the complexity of the misère version of three games played on graphs. Theoret. Comput. Sci. 595 159–167.
• [17] Schaefer, T. J. (1978). On the complexity of some two-person perfect-information games. J. Comput. System Sci. 16 185–225.
• [18] Schonmann, R. H. (1992). On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20 174–193.
• [19] Spencer, J. (2001). The Strange Logic of Random Graphs. Algorithms and Combinatorics 22. Springer, Berlin.
• [20] Wästlund, J. (2012). Replica symmetry of the minimum matching. Ann. of Math. (2) 175 1061–1091.