The Annals of Applied Probability

Trapping games on random boards

Riddhipratim Basu, Alexander E. Holroyd, James B. Martin, and Johan Wästlund

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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
Received: May 2015
Revised: February 2016
First available in Project Euclid: 15 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1481792598

Digital Object Identifier
doi:10.1214/16-AAP1190

Mathematical Reviews number (MathSciNet)
MR3582816

Zentralblatt MATH identifier
1356.05088

Subjects
Primary: 05C57: Games on graphs [See also 91A43, 91A46] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C70: Factorization, matching, partitioning, covering and packing

Keywords
Combinatorial game percolation maximum matching maximum independent set boundary conditions

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


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