AI 中文总结
本文研究有线边界条件下Δ-正则树上 $q<1$ 的随机团模型，证明超临界有线DLR规范有唯一吉布斯测度，建立有线分支间的负相关性及相关协方差不等式。
AI 中文摘要
具有团权重 $0<q<1$ 的随机团模型被认为会呈现负相关性，但由于缺乏FKG不等式，即使是一般图上的成对负相关性问题仍未解决。我们研究该模型在带有有线边界条件的无限Δ-正则树上的情况。记 $\u0070\u0196:=\u0070/(\u0070+q(1-\u0070))$，$\u0070_{\text{c}}:=q/(\u0394+q-2)$。经典结果确定了当 $\u0070\u2264\u0070_{\text{c}}$ 时的乘积有线态，并构造了当 $\u0070>\u0070_{\text{c}}$ 时的渗流全有线极限。我们证明超临界有线DLR规范具有唯一的吉布斯测度，即该全有线极限。因此，有线DLR相图是完整的：当 $\u0070\u2264\u0070_{\text{c}}$ 时，唯一测度是参数为 $\u0070\u0196$ 的伯努利键渗流；当 $\u0070>\u0070_{\text{c}}$ 时，它会发生渗流。我们还建立了有线分支间的负相关性。在有限有线树上，分支连通性指标向量在正外场下满足条件负关联（CNA+）。因此，支撑于同一顶点关联的不相交分支集合上的有界递增可观测量具有非正协方差。对于任意 $\u0070$，相同不等式在唯一的无限体积有线测度中成立，当 $\u0070\u2264\u0070_{\text{c}}$ 时等号成立。
英文摘要
The random-cluster model with cluster weight $0<q<1$ is expected to exhibit negative dependence, but without FKG even pairwise negative correlation remains open on general graphs. We study the model on the infinite $Δ$-regular tree with wired boundary conditions. Write $\widehat p:=p/(p+q(1-p))$ and $p_{\mathsf c}:=q/(Δ+q-2)$. Classical results identify the product wired state for $p\le p_{\mathsf c}$ and construct a percolative all-wired limit for $p>p_{\mathsf c}$. We prove that the supercritical wired DLR specification has a unique Gibbs measure, namely this all-wired limit. Consequently, the wired DLR phase diagram is complete: the unique measure is Bernoulli bond percolation with parameter $\widehat p$ for $p\le p_{\mathsf c}$, while it percolates for $p>p_{\mathsf c}$. We also establish negative dependence across wired branches. On a finite wired tree, the vector of branch-connectivity indicators satisfies conditional negative association under positive external fields (CNA+). Hence bounded increasing observables supported on disjoint collections of branches incident to a common vertex have nonpositive covariance. The same inequality holds in the unique infinite-volume wired measure for every $p$, with equality for $p\le p_{\mathsf c}$.
Comments32 pages, 1 figure
