一类无限族吸附模型和限制Lukasiewicz路径
An infinite family of adsorption models and restricted Lukasiewicz paths
摘要：
我们定义了$(k,\ell)$限制的Lukasiewicz路径，$k\le\ell\in\mathbb{N}_0$，并使用这些路径作为模型的聚合物吸附。我们写下一个多项式表达式满足的生成函数的任意值的$(k,\ell)$。所得多项式的次数为$\ell+1$，因此对于足够大的$\ell$不能显式求解。我们提供了两种不同的方法来获得相图。除了一个更传统的分析，我们还开发了一个新的数学表征的相图中的多项式和零的最高程度的系数的判别式。然后，我们给出了一个双射之间的$(k,\ell)$ -限制Lukasiewicz路径和“上升”-限制戴克路径，确定另一个家庭的路径模型共享相同的关键行为。对于$(k,\ell)=(1,\infty)$，我们提供了一个新的Motzkin路的双射。我们还考虑了面积加权生成函数，并证明它是一个q-变形代数函数。我们确定的生成函数明确在特定情况下的$(k,\ell)$限制Lukasiewicz路径，并为$(k,\ell)=(0,\infty)$，我们提供了一个双射Dyck路径。
英文摘要：
We define $(k,\ell)$-restricted Lukasiewicz paths, $k\le\ell\in\mathbb{N}_0$, and use these paths as models of polymer adsorption. We write down a polynomial expression satisfied by the generating function for arbitrary values of $(k,\ell)$. The resulting polynomial is of degree $\ell+1$ and hence cannot be solved explicitly for sufficiently large $\ell$. We provide two different approaches to obtain the phase diagram. In addition to a more conventional analysis, we also develop a new mathematical characterization of the phase diagram in terms of the discriminant of the polynomial and a zero of its highest degree coefficient. We then give a bijection between $(k,\ell)$-restricted Lukasiewicz paths and "rise"-restricted Dyck paths, identifying another family of path models which share the same critical behaviour. For $(k,\ell)=(1,\infty)$ we provide a new bijection to Motzkin paths. We also consider the area-weighted generating function and show that it is a q-deformed algebraic function. We determine the generating function explicitly in particular cases of $(k,\ell)$-restricted Lukasiewicz paths, and for $(k,\ell)=(0,\infty)$ we provide a bijection to Dyck paths.
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