具有$|x-y|^{-2}$相互作用的Dyson分层模型中的双指数易感性增长
Double-exponential susceptibility growth in Dyson's hierarchical model with $|x-y|^{-2}$ interaction
摘要：
本文研究了$d$维等级格点上的长程渗流问题，其中每个可能的边$\{x,y\}$以包含概率$1-\exp ( -β\|x-y\|^{-d-α} )$独立地随机包含，其中$α>0$是固定的，$β\geq 0$是一个参数.已知该模型在某个$β_c<\infty$处有相变，当且仅当$α<d$。我们研究了$α\geq d$体系中的模型，其中$β_c=\infty$，并证明了磁化率$χ(β)$（即，这解决了Georgakopoulos和Haslegrave（2020）提出的一个问题，他们表明当$α=d$时，$χ(β)$在指数和双指数之间增长。我们的研究结果意味着，类似的结果持有一些相关的模型，包括戴森的分层伊辛模型，我们建立的双指数易感性增长似乎是一个新的现象，即使在启发式水平。
英文摘要：
We study long-range percolation on the $d$-dimensional hierarchical lattice, in which each possible edge $\{x,y\}$ is included independently at random with inclusion probability $1-\exp ( -β\|x-y\|^{-d-α} )$, where $α>0$ is fixed and $β\geq 0$ is a parameter. This model is known to have a phase transition at some $β_c<\infty$ if and only if $α<d$. We study the model in the regime $α\geq d$, in which $β_c=\infty$, and prove that the susceptibility $χ(β)$ (i.e., the expected volume of the cluster at the origin) satisfies \[ χ(β) = β^{\frac{d}{α- d } - o(1)} \qquad \text{as $β\to \infty$ if $α> d$} \qquad \text{and} \qquad e^{e^{ Θ(β) }} \qquad \text{as $β\to \infty$ if $α= d$.} \] This resolves a problem raised by Georgakopoulos and Haslegrave (2020), who showed that $χ(β)$ grows between exponentially and double-exponentially when $α=d$. Our results imply that analogous results hold for a number of related models including Dyson's hierarchical Ising model, for which the double-exponential susceptibility growth we establish appears to be a new phenomenon even at the heuristic level.
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