Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with 1 ≤ q ≤ 4
Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with 1 ≤ q ≤ 4
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DOI：
10.1007/s00220-016-2759-8
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发表时间：
2017-01-01
影响因子：
2.4
通讯作者：
Tassion, Vincent
中科院分区：
文献类型：
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作者：
Duminil-Copin, Hugo;Sidoravicius, Vladas;Tassion, Vincent
本文研究了平面Potts模型及其随机簇表示。我们证明了在单色边界条件下，最近邻铁磁q态Potts模型的相变是连续的，即存在唯一的Gibbs态，或者等价地说临界Gibbs态没有序，证明使用了带团簇权的随机团簇模型（注意q不一定是整数），并且基于两个成分：自由状态的两点函数对于簇权重以亚指数快速衰减的事实，该结果是在研究离散黎曼曲面上的仿费米观测量时得到的，证明了临界随机集团模型的几个性质的等价性：线边界条件下无限簇的不存在，无限体积测度的唯一性，自由边界条件下两点函数的次指数衰减，本文给出了矩形区域内任意边界条件下交叉概率的Russo-Seymour-Welsh型结果，该结果对研究随机集团模型的标度极限具有重要意义。它表明，当采用缩放极限时，界面族（例如Dobrushin边界条件）是紧的，并且任何子序列极限都可以通过Loewner链参数化。我们还研究了边界条件对这些子序列极限的影响。让我们提一下，这个结果在临界指数的研究中也应该是有用的。
This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model on is continuous for , in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions.The proof uses the random-cluster model with cluster-weight (note that q is not necessarily an integer) and is based on two ingredients: The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights , which is derived studying parafermionic observables on a discrete Riemann surface.A new result proving the equivalence of several properties of critical random-cluster models: he absence of infinite-cluster for wired boundary conditions,the uniqueness of infinite-volume measures,the sub-exponential decay of the two-point function for free boundary conditions,a Russo-Seymour-Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions.The result has important consequences toward the study of the scaling limit of the random-cluster model with . It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.