论文标题
SOS模型和有限旋转模型的极端不均匀吉布斯状态
Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees
论文作者
论文摘要
我们考虑$ \ mathbb z $ - 价值$ p $ -sos -sos-Models与形式$ |ω_v-ω_w|^p $最近的邻居相互作用，以及常规树上有限的旋转铁磁模型。这包括经典的SOS模型，离散的高斯模型和Potts模型。我们表现出一个极端不均匀的家族（即树自动形态不变）Gibbs，其测量是基态的低温扰动（局部能量最小化器），它们具有足够稀疏的一组破碎的键，以及沿它们沿它们均匀边界的增量。这些低温状态通常不具有树的任何对称性。这概括了Gandolfo，Ruiz和Shlosman \ cite {grs12}关于Ising模型的结果，并表明后一种行为是强大的。我们处理三种不同类型的扩展：非紧凑状态空间梯度模型，无自旋对称的模型以及小型随机场中的模型。我们提供了所有Gibbs测量中低温状态的极端性的详细构造和完整的证明，分析相对于基态的过量能量，低温扩张的收敛以及切割的特性。
We consider $\mathbb Z$-valued $p$-SOS-models with nearest neighbor interactions of the form $|ω_v-ω_w|^p$, and finite-spin ferromagnetic models on regular trees. This includes the classical SOS-model, the discrete Gaussian model and the Potts model. We exhibit a family of extremal inhomogeneous (i.e. tree automorphism non-invariant) Gibbs measures arising as low temperature perturbations of ground states (local energy minimizers), which have a sparse enough set of broken bonds together with uniformly bounded increments along them. These low temperature states in general do not possess any symmetries of the tree. This generalises the results of Gandolfo, Ruiz and Shlosman \cite{GRS12} about the Ising model, and shows that the latter behaviour is robust. We treat three different types of extensions: non-compact state space gradient models, models without spin-symmetry, and models in small random fields. We give a detailed construction and full proofs of the extremality of the low-temperature states in the set of all Gibbs measures, analysing excess energies relative to the ground states, convergence of low-temperature expansions, and properties of cutsets.
