Matching Rectangles in d - Dimensions : Algorithms and Laws of Large Numbers

For each point of the integer lattice Zd, let X and Y be independent identically distributed random variables with P(X = Y) = p E (0, 1). Let S(n) be the volume of the largest d-dimensional cube in ( I , ..., n t d with the property that X = Y at every point of the cube; R ( n ) is similarly defined to be the maximum volume of perfectly matching rectangles. It is proved that, if all possible shifts of the X lattice relative to the Y lattice are allowed, P(limm-,m S(n)/log n = limn+m R(n)/ log n = 2 4 = 1, where log is to base (l/p). The corresponding limit without shifts is d. Algorithms to find largest squares and rectangles, with and without shifts, are also given. @ 198s Academic P n s . Inc.