论文标题
伊辛相变的连续性在不合同的基团上
Continuity of the Ising phase transition on nonamenable groups
论文作者
论文摘要
我们严格地证明，任何不合同的cayley图上的铁磁ising模型都会发生连续的（二阶）相变，即在临界温度下有独特的吉布斯度量。该定理的证明是定量的，并且在临界和接近临界的磁化方面也产生了幂律界限。的确，我们更广泛地证明，磁化$ \langleσ_o\ rangle_ {β，h}^+$是反向温度$β$和外部字段$β$和外部字段$ h $的本地hölder-contiul函数，整个非负量象限（β，h）\ in [0，\ infty，\ infty）^infty，\ infty，\ infty）^2 $^2 $^2 $。作为我们开发的方法的第二次应用，我们还证明，在任何瞬时不可符号图上，Bernoulli Percolation的自由能在$ p_c $时是两倍。
We prove rigorously that the ferromagnetic Ising model on any nonamenable Cayley graph undergoes a continuous (second-order) phase transition in the sense that there is a unique Gibbs measure at the critical temperature. The proof of this theorem is quantitative and also yields power-law bounds on the magnetization at and near criticality. Indeed, we prove more generally that the magnetization $\langle σ_o \rangle_{β,h}^+$ is a locally Hölder-continuous function of the inverse temperature $β$ and external field $h$ throughout the non-negative quadrant $(β,h)\in [0,\infty)^2$. As a second application of the methods we develop, we also prove that the free energy of Bernoulli percolation is twice differentiable at $p_c$ on any transitive nonamenable graph.
