标题： 伊辛模型中的相变：衰减场下的半无限情况和长程随机场 显示英文标题

标题： Phase Transitions in Ising models: the Semi-infinite with decaying field and the Random Field Long-range

摘要： 在本论文中，我们提出了关于两个模型相变的结果：具有衰减场的半无限伊辛模型，以及具有随机场的长程伊辛模型。 我们研究了具有外场$h_i = \lambda |i_d|^{-\delta}$的半无限伊辛模型，$\lambda$是壁的影响，以及$\delta>0$。 这个外场随着远离壁而衰减。 我们能够证明当$\delta>1$和$\beta > \beta_c(d)$时，存在一个临界值$0< \lambda_c:=\lambda_c(\delta,\beta)$，使得对于$\lambda<\lambda_c$存在相变，而对于$\lambda>\lambda_c$我们有吉布斯态的唯一性。 此外，当 $\delta<1$时，对于任何正的 $\beta$ 和 $\lambda$，我们只有一个吉布斯态。 对于具有随机场的模型，我们将丁和庄最近从最近邻到长程相互作用的论证进行了扩展，并证明了铁磁随机场伊辛模型类中的相变。 我们的证明结合了阿方索、比萨科特、恩多和汉达提出的多维设置下弗罗利希-斯宾塞轮廓的推广，以及费舍尔、弗罗利希和斯宾塞引入的粗粒化过程。 我们的结果表明，如果存在合适的轮廓系统，在维度 $d\geq 3$ 中当 $\alpha > d$ 时，丁-庄策略对于相互作用 $J_{xy}=|x-y|^{- \alpha}$ 也是有用的，从而提供了一种不使用重整化群方法（RGM）的替代证明，因为布里孔和库皮埃恩声称RGM也应该适用于这种普遍性。 我们可以考虑独立同分布的随机场，其服从高斯分布或伯努利分布。 显示英文摘要

摘要： In this thesis, we present results on phase transition for two models: the semi-infinite Ising model with a decaying field, and the long-range Ising model with a random field. We study the semi-infinite Ising model with an external field $h_i = \lambda |i_d|^{-\delta}$, $\lambda$ is the wall influence, and $\delta>0$. This external field decays as it gets further away from the wall. We are able to show that when $\delta>1$ and $\beta > \beta_c(d)$, there exists a critical value $0< \lambda_c:=\lambda_c(\delta,\beta)$ such that, for $\lambda<\lambda_c$ there is phase transition and for $\lambda>\lambda_c$ we have uniqueness of the Gibbs state. In addition, when $\delta<1$ we have only one Gibbs state for any positive $\beta$ and $\lambda$. For the model with a random field, we extend the recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in the class of ferromagnetic random field Ising models. Our proof combines a generalization of Fr\"ohlich-Spencer contours to the multidimensional setting proposed by Affonso, Bissacot, Endo and Handa, with the coarse-graining procedure introduced by Fisher, Fr\"ohlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions $J_{xy}=|x-y|^{- \alpha}$ when $\alpha > d$ in dimension $d\geq 3$ if we have a suitable system of contours, yielding an alternative proof that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen claimed that the RGM should also work on this generality. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions.