Cayley树上的硬核和软核Widom-Rowlinson模型
Hard-Core and Soft-Core Widom-Rowlinson models on Cayley trees
摘要：
我们考虑硬核和软核Widom-Rowlinson模型的自旋值$-1,0,1$的Cayley树顺序$k\geq 2$，我们感兴趣的吉布斯措施的模型。模型依赖于3个参数：树的顺序$k$，描述（铁磁或反铁磁）相互作用强度的$θ$，以及描述粒子强度的$λ$。硬核Widom-Rowlinson模型对应于$θ=0$案例。对于二叉树$k=2$和$k=3$，我们证明了铁磁模型有一个或三个分裂吉布斯测度（树自同构不变吉布斯测度（TISGM）是树索引的马尔可夫链）。我们还给出了相应的临界曲线在参数空间中的精确形式。对于更高的值$k$，我们给出了一个明确的充分界，确保非唯一性，我们猜想是确切的曲线。此外，对于反铁磁模型，我们明确地给出了两条临界曲线，并证明了在这些曲线上恰好有两个TISGM;在这些曲线之间恰好有三个TISGM;否则存在唯一的TISGM。在铁磁模型中还构造了一些周期和非周期的自稳磁模型。
英文摘要：
We consider both Hard-Core and Soft-Core Widom-Rowlinson models with spin values $-1,0,1$ on a Cayley tree of order $k\geq 2$ and we are interested in the Gibbs measures of the models. The models depend on 3 parameters: the order $k$ of the tree, $θ$ describing the strength of the (ferromagnetic or antiferromagnetic) interaction, and $λ$ describing the intensity for particles. The Hard-Core Widom-Rowlinson model corresponds to the case $θ=0$. For the binary tree $k=2$, and for $k=3$ we prove that the ferromagnetic model has either one or three splitting Gibbs measures (tree-automorphism invariant Gibbs measures (TISGM) which are tree-indexed Markov chains). We also give the exact form of the corresponding critical curves in parameter space. For higher values of $k$ we give an explicit sufficient bound ensuring non-uniqueness which we conjecture to be the exact curve. Moreover, for the antiferromagnetic model we explicitly give two critical curves and prove that on these curves there are exactly two TISGMs; between these curves there are exactly three TISGMs; otherwise there exists a unique TISGM. Also some periodic and non-periodic SGMs are constructed in the ferromagnetic model.
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