论文标题
3D堆的增长率
Growth rate of 3D heaps of pieces
论文作者
论文摘要
我们考虑在一个简单的三维件（$ n \ gg 1 $）的三维量的配置统计量，该量子在一个简单的立方晶格上，在一个大的3D框架基本$ n \ times n $的3D边界盒中，并计算出相应分区的$λ（n）$的增长率，相应的分区功能，$ z_n \ sim n^usn^us n^usim n^us $ n $ n $ n $ n $ n $ n）$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $。我们的计算依赖于G.X的定理。 viennot \ cite {viennot-rev}，它连接了$（d+1）$ - 尺寸件的生成函数与这些零件投影到$ d $维二维子空间的生成函数的生成函数。因此，一堆立方块的生长速率无法通过垂直面相互接触，因此与描述硬正方形的2D晶格气体的零位置有关。我们在有限的$ n \ times n $ lattice of nutary $ n $的有限密度上精确地研究了相应的分区功能，并将其行为推断到干扰过渡密度。这使我们能够估算限制增长率，$λ= \ lim_ {n \ to \ infty}λ（n）\约9.5 $。同样的方法适用于任何基础2D晶格和各种形状的零件：平坦的垂直正方形，映射到排斥二聚体的合奏，多米诺骨牌映射到具有硬核排斥等矩形的集合等。
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, $Λ(n)$, of the corresponding partition function, $Z_N\sim N^θ[Λ(n)]^N$, at $n\gg 1$. Our computations rely on a theorem of G.X. Viennot \cite{viennot-rev}, 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, $Λ=\lim_{n\to\infty}Λ(n)\approx 9.5$. The same method works for any 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.
