We obtain limit shape of perfect matchings on a large class of rail-yard graphs with right boundary condition given by the empty partition and left boundary condition given by either by a staircase partition with constant density or a piecewise partition with densities either 1 or 0. We prove the parametric equations for the frozen boundary and find conditions under which the frozen boundary is a cloud curve, or a union of disjoint cloud curves.

中文翻译：

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Rail-Yard 图上完美匹配的极限形状

我们在一大类铁路场图上获得完美匹配的极限形状，右边界条件由空分区给出，左边界条件由恒定密度的楼梯分区或密度为 1 或 0 的分段分区给出。我们证明了冻结边界的参数方程，并找到了冻结边界为云曲线或不相交云曲线的联合的条件。