We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a weight \(r>0\). We give a variational principle for limit shapes in this setting, and show that the resulting Euler–Lagrange equation can be integrated, giving limit shapes explicitly parameterized by analytic functions.

中文翻译：

---

非对称五顶点模型的限制形状

我们计算了五顶点模型的自由能和表面张力函数，这是网格上不相交单调晶格路径的模型，其中每个角都有一个权重\(r>0\)。我们在此设置中给出了极限形状的变分原理，并表明所得的欧拉-拉格朗日方程可以积分，给出由解析函数明确参数化的极限形状。