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Random tilings of geometrical shapes with dominos or lozenges have been arich source of universal statistical distributions. This paper deals withdomino tilings of checker board rectangular shapes such that the top two andbottom two adjacent squares have the same orientation and the two most left andtwo most right ones as well.  It forces these so-called "skew-Aztec rectangles" to have cuts on eitherside. For large sizes of the domain and upon an appropriate scaling of thelocation of the cuts, one finds split tacnodes between liquid regions with twodistinct adjacent frozen phases descending into the tacnode. Zooming about suchsplit tacnodes, filaments appear between the liquid patches evolving in abricklike sea of dimers of another type. This work shows that the randomfluctuations in a neighborhood of the split tacnode are governed asymptoticallyby the discrete tacnode kernel, providing strong evidence that this kernel is auniversal discrete-continuous limiting kernel occurring naturally whenever wehave double interlacing patterns. The analysis involves the inversion of asingular Toeplitz matrix which leads to considerable difficulties.

A singular Toeplitz determinant and the discrete tacnode kernel for skew-Aztec Rectangles