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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 correspondingpartition function, $Z_N\sim N^{\theta}[\Lambda(n)]^N$, at $n\gg 1$. Ourcomputations rely on a theorem of G.X. Viennot \cite{viennot-rev}, whichconnects the generating function of a $(D+1)$-dimensional heap of pieces to thegenerating function of projection of these pieces onto a $D$-dimensionalsubspace. The growth rate of a heap of cubic blocks, which cannot touch eachother by vertical faces, is thus related to the position of zeros of thepartition function describing 2D lattice gas of hard squares. We study thecorresponding partition function exactly at low densities on finite $n\times n$lattice of arbitrary $n$, and extrapolate its behavior to the jammingtransition density. This allows us to estimate the limiting growth rate,$\Lambda =\lim_{n\to\infty}\Lambda(n)\approx 9.5$. The same method works forany underlying 2D lattice and for various shapes of pieces: flat verticalsquares, mapped to an ensemble of repulsive dimers, dominoes mapped to anensemble of rectangles with hard-core repulsion, etc.

Growth rate of 3D heaps of pieces