Many dimensions. In d dimensions the lattice points i = (i1,…,id) where 1 ≤ ij ≤ n may…

Many dimensions. In d dimensions the lattice points i = (i1,…,id) where 1 ≤ ij ≤ n may…

Many dimensions. In d dimensions the lattice points i = (i1,…,id) where 1 ≤ ij ≤ n may be identified with the “squares” (or, better perhaps, “d-dimensional unit cuboids”) of a d-dimensional chessboard ranging over n cuboids in each dimension. A d-dimensional rook located at the lattice point (i1,…,id) can range freely along points in directions parallel to the coordinate axes (varying ij , for instance, while keeping ik for k = j fixed). Suppose r rooks are placed at random on the d-dimensional chessboard. Say that a given rook is “secure” if it does not lie along the axis-parallel lines of sight of any of the other r−1 rooks. Show that the number of secure rooks satisfies an asymptotic Poisson law for a critical rate of growth of r = rn with n

 

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