Open Access
December 2016 Trapping games on random boards
Riddhipratim Basu, Alexander E. Holroyd, James B. Martin, Johan Wästlund
Ann. Appl. Probab. 26(6): 3727-3753 (December 2016). DOI: 10.1214/16-AAP1190


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.


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Riddhipratim Basu. Alexander E. Holroyd. James B. Martin. Johan Wästlund. "Trapping games on random boards." Ann. Appl. Probab. 26 (6) 3727 - 3753, December 2016.


Received: 1 May 2015; Revised: 1 February 2016; Published: December 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1356.05088
MathSciNet: MR3582816
Digital Object Identifier: 10.1214/16-AAP1190

Primary: 05C57 , 05C70 , 60K35

Keywords: boundary conditions , Combinatorial game , maximum independent set , maximum matching , percolation

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 6 • December 2016
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