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I've seen the following remark in a number of papers but don't know what to make of it. In this paper by Cohn, Elkies and Propp, it is mentioned that the normalized average Height function $\mathcal{H}(x,y)$ for the Aztec Diamond is related to the wave equation. On page 26 of the above paper, the equations for the temperate zone of the Aztec Diamond reduce to

$$\frac{\partial^2\mathcal{H}}{\partial y^2}-\frac{\partial^2\mathcal{H}}{\partial x^2}=\frac{8}{\pi\sqrt{1-2x^2-2y^2}}$$

and if one now rescales $\overline{\mathcal{H}}(x,y,t):=t\mathcal{H}(x/t,y/t)$, then

$$\frac{\partial^2\overline{\mathcal{H}}}{\partial y^2}-\frac{\partial^2\overline{\mathcal{H}}}{\partial x^2}=8u(x,y,t),$$

where $u(x,y,t)=1/\pi\sqrt{t^2-2x^2-2y^2}$. In other words, $u$ is the fundamental solution to the wave equation in 2D with propagation speed $1/\sqrt{2}$, e.g. $u_t=\frac{1}{2}(u_{xx}+u_{yy})$.

Further remarks in the above paper mention that it is possible to use the wave equation to model creation rates (which is related to the domino shuffling algorithm). Moreover, recent work by Francesco and Soto-Garrido has shown that the densities $u$ of dominos can be rederived through the fact that the partition function for dominos satisfies an Octohedral recurrence.

Questions: Is there a holistic explanation for why $u$ should satisfy the wave equation? In particular, what exactly does PDE theory have to say about arctic circle limit shapes? Have there been more generalized results of this PDE nature corresponding to other kinds of tilings? Ultimately, is there something deeper going on here than just some elegant-looking asymptotic generating function relations?

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