论文标题
与随机电导的谐波晶体的断开连接和熵排斥
Disconnection and entropic repulsion for the harmonic crystal with random conductances
论文作者
论文摘要
我们研究了$ \ Mathbb {z}^d $，$ d \ geq 3 $的谐波晶体的级别渗透，并具有均匀的椭圆形随机电导。我们证明，该模型在环境度量下几乎肯定是恒定的，在临界水平上进行了非平凡的相变。此外，我们研究了断开事件，即该字段以下$α$的级别集可以断开紧凑型集合$ a \ subseteq \ subseteq \ mathbb {r}^d $的离散连接。利用Neukamm，Schäffner和Schlömerkemper的结果，我们从$ a $ a $ a的均质能力方面获得了淬灭的渐近上限和下限，请参见Arxiv：1606.06533。此外，我们给出了上限的概率，即在断开连接时，该场的局部平均值会根据$ a $ a偏离某些配置文件功能的可能性。关于我们得出的断开连接的上限和下边界在领先顺序上可能匹配。在这种情况下，这项工作表明，对断开连接的调节会导致该场的熵推入。本文的结果概括了Arxiv的发现：1802.02518和Arxiv：1808.09947由作者处理恒定电导的情况。我们的证明涉及随机步行的新颖“固化估计值”，这本质上与Sznitman衍生的布朗运动的相应估计值和Arxiv中的第二位作者相似：1706.07229。
We study level-set percolation for the harmonic crystal on $\mathbb{Z}^d$, $d \geq 3$, with uniformly elliptic random conductances. We prove that this model undergoes a non-trivial phase transition at a critical level that is almost surely constant under the environment measure. Moreover, we study the disconnection event that the level-set of this field below a level $α$ disconnects the discrete blow-up of a compact set $A \subseteq \mathbb{R}^d$ from the boundary of an enclosing box. We obtain quenched asymptotic upper and lower bounds on its probability in terms of the homogenized capacity of $A$, utilizing results from Neukamm, Schäffner and Schlömerkemper, see arXiv:1606.06533. Furthermore, we give upper bounds on the probability that a local average of the field deviates from some profile function depending on $A$, when disconnection occurs. The upper and lower bounds concerning disconnection that we derive are plausibly matching at leading order. In this case, this work shows that conditioning on disconnection leads to an entropic push-down of the field. The results in this article generalize the findings of arXiv:1802.02518 and arXiv:1808.09947 by the authors which treat the case of constant conductances. Our proofs involve novel "solidification estimates" for random walks, which are similar in nature to the corresponding estimates for Brownian motion derived by Sznitman and the second author in arXiv:1706.07229.
