无序金刚石晶格上定向聚合物的嵌套临界点
Nested critical points for a directed polymer on a disordered diamond lattice
摘要：
我们考虑一个模型的定向聚合物在一个随机环境中定义的分层钻石晶格，其中i.i.d.随机变量附着在晶格键上。我们的重点是缩放方案，其中的尺寸参数$n$，计数系统的层次层的数量，成为大的逆温度$β$消失。当$β$具有参数$\widehatβ>0$的形式$\widehatβ/\sqrt{n}$时，我们证明了存在一个截止值$0 < κ< \infty$，使得作为$n \to \infty$，归一化配分函数的方差对于$\widehatβ\leq κ$趋于零，并且对于$\widehatβ > κ$无约束地增长。我们通过将逆温度设置为$κ/\sqrt{n} + α_n$（其中$0 < α_n \ll 1/\sqrt{n}$）并分析方差的渐近行为，获得了这两种制度之间边界的更精确描述。我们表明，当$α_n = α(\log n-\log \log n)/n^{3/2}$（与一个小的修改，以处理非零的三阶矩）有一个类似的截止值$η$参数$α$，使当$α< η$的方差为零，增长没有约束时，$α> η$。通过探测逆温度$κ/\sqrt{n} + η(\log n-\log \log n)/n^{3/2}$再次扩展分析，我们发现归一化配分函数的方差行为的嵌套临界点的无限序列。在亚临界情况下，$\widehatβ \leq κ$和$α\leq η$这种分析扩展到一个中心极限定理结果的波动的归一化配分函数。
英文摘要：
We consider a model for a directed polymer in a random environment defined on a hierarchical diamond lattice in which i.i.d. random variables are attached to the lattice bonds. Our focus is on scaling schemes in which a size parameter $n$, counting the number of hierarchical layers of the system, becomes large as the inverse temperature $β$ vanishes. When $β$ has the form $\widehatβ/\sqrt{n}$ for a parameter $\widehatβ>0$, we show that there is a cutoff value $0 < κ< \infty$ such that as $n \to \infty$ the variance of the normalized partition function tends to zero for $\widehatβ\leq κ$ and grows without bound for $\widehatβ > κ$. We obtain a more refined description of the border between these two regimes by setting the inverse temperature to $κ/\sqrt{n} + α_n$ where $0 < α_n \ll 1/\sqrt{n}$ and analyzing the asymptotic behavior of the variance. We show that when $α_n = α(\log n-\log \log n)/n^{3/2}$ (with a small modification to deal with non-zero third moment) there is a similar cutoff value $η$ for the parameter $α$ such that when $α< η$ the variance goes to zero and grows without bound when $α> η$. Extending the analysis yet again by probing around the inverse temperature $κ/\sqrt{n} + η(\log n-\log \log n)/n^{3/2}$ we find an infinite sequence of nested critical points for the variance behavior of the normalized partition function. In the subcritical cases $\widehatβ \leq κ$ and $α\leq η$ this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.
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