Growth rate of 3D Tetris

We consider configurational statistics of three-dimensional heaps of $N$ pieces ($N\gg 1$) on a simple cubic lattice in a large 3D bounding box of base $n \times n$, and calculate the growth rate, $\Lambda(n)$, of the corresponding partition function, $Z_N\sim N^{\theta}[\Lambda(n)]^N$, at $n\gg 1$. Our computations rely on a theorem of G.X. Viennot, which connects the generating function of a $(D+1)$-dimensional heap of pieces to the generating function of projection of these pieces onto a $D$-dimensional subspace. The growth rate of a heap of cubic blocks, which cannot touch each other by vertical faces, is thus related to the position of zeros of the partition function describing 2D lattice gas of hard squares. We study the corresponding partition function exactly at low densities on finite $n\times n$ lattice of arbitrary $n$, and extrapolate its behavior to the jamming transition density. This allows us to estimate the limiting growth rate, $\Lambda =\lim_{n\to\infty}\Lambda(n)\approx 9.5$. The same method works for any graph above the underlying 2D lattice and for various shapes of pieces: flat vertical squares, mapped to an ensemble of repulsive dimers, dominoes mapped to an ensemble of rectangles with hard-core repulsion, etc.