To expand slightly on Joseph's answer and Noam's comment, you can describe a domino or lozenge tiling by its "height function." In the case of lozenge tilings, shown in Joseph's answer, the height function is visible, since the picture can naturally be interpreted as showing a 3d surface. (You have to make one choice, since there's an ambiguity regarding what is pointing into the screen and what is pointing out.) For domino tilings, it is not nearly as visually apparent, but height functions are explained in a beautiful paper by Thurston (Conway's tiling groups, Amer. Math. Monthly 97 (1990), 757–773).
The microscopic shape of a region influences the height function, since it determines the wire-frame boundary the height function must interpolate. For example, for Aztec diamonds the boundary is as steep as possible, while in the coarser example it stays flat.
It turns out that in the continuum limit, the shape of a typical tiling is entirely determined by the shape of the boundary, and it is determined by a variational principle: among all spanning surfaces that satisfy a certain Lipschitz constraint (that must hold for all tilings), the typical surface is singled out by maximizing the entropy, which is given by integrating local contributions that can be computed explicitly in terms of the partial derivatives of the height function. See http://front.math.ucdavis.edu/math.CO/0008220 for more details. Almost all tilings will cluster around the typical tiling, so this completely determines the large-scale behavior.
The two examples from the question can be completely analyzed in this framework. In particular, the second one has the highest possible entropy at every point and therefore has nothing like an Arctic circle. See Figure 4 in http://front.math.ucdavis.edu/math.CO/0008243 (reproduced below) for a picture of a random tiling.
Kenyon, Okounkov, and Sheffield (http://front.math.ucdavis.edu/0311.4705) substantially generalized this framework, and Kenyon and Okounkov (http://front.math.ucdavis.edu/0507.4707) found a beautiful description of the variational problem for lozenge tilings in terms of the complex Burgers equation. This leads to explicit descriptions of cases like the cardioid in Joseph's answer.