Lambda-determinants and domino-tilings

Consider the 2n-by-2n matrix M=(mi,j)i,j=12n with mi,j=1 for i,j satisfying |2i−2n−1|+|2j−2n−1|⩽2n and mi,j=0 for all other i,j, consisting of a central diamond of 1's surrounded by 0's. When n⩾4, the λ-determinant of the matrix M (as introduced by Robbins and Rumsey [Adv. Math. 62 (1986) 169-184]) is not well defined. However, if we replace the 0's by t's, we get a matrix whose λ-determinant is well defined and is a polynomial in λ and t. The limit of this polynomial as t→0 is a polynomial in λ whose value at λ=1 is the number of domino-tilings of a 2n-by-2n square.