Trapping games on random boards. (English) Zbl 1356.05088

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

Ann. Appl. Probab. 26, No. 6, 3727-3753 (2016).

Summary: 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 \(d\geq 3\). 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.

MSC:

05C57	Games on graphs (graph-theoretic aspects)
05C69	Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C35	Extremal problems in graph theory
91A43	Games involving graphs
91A05	2-person games
91A46	Combinatorial games
60K35	Interacting random processes; statistical mechanics type models; percolation theory
05C70	Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)

Keywords:

combinatorial game; percolation; maximum matching; maximum independent set; boundary conditions