arXiv 1508.047912015-08-19math.PR

等级格点上有向聚合物模型的中间无序区

The intermediate disorder regime for a directed polymer model on a hierarchical lattice

Tom Alberts, Jeremy Clark, Sasa Kocic

备注：: 41 pages, 2 figures

摘要：

我们研究了一个有向聚合物模型上定义的分级钻石晶格，其中的晶格是递归地构造通过一个配方取决于一个分支数$b\in \mathbb{N}$和一个片段数$s\in \mathbb{N}$。当$b\leq s$以前的工作[27]已经确定该模型对逆温度$β$的所有正值都表现出强无序时，因此弱无序仅对$β=0$（无限温度）起作用。我们的重点是在所谓的中间无序制度，其中的逆温度$β\equiv β_{n}$消失在一个适当的速率作为系统的大小$n$的增长。我们的分析需要对$b<s$和$b=s$病例进行单独治疗。在$b<s$的情况下，我们证明，当逆温度采取的形式$β_{n}=\widehatβ (b/s)^{n/2}$的$\widehatβ>0$，系统的归一化配分函数收敛弱$n \to \infty$的分布$\mathbf{L}(\widehatβ)$连续依赖于参数$\widehatβ$。在$b=s$的情况下，我们发现一个临界点的行为时，逆温度的模型是缩放为$β_{n}=\widehatβ/n$ ;对于一个明确的可计算的临界值$κ_{b} > 0$的归一化配分函数的方差收敛到零与大$n$时，$\widehatβ\leq κ_{b}$和增长没有约束时，$\widehatβ>κ_{b}$。最后证明了当$\widehatβ\leq κ_{b}$ .

英文摘要：

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number $b\in \mathbb{N}$ and a segment number $s\in \mathbb{N}$. When $b\leq s$ previous work [27] has established that the model exhibits strong disorder for all positive values of the inverse temperature $β$, and thus weak disorder reigns only for $β=0$ (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature $β\equiv β_{n}$ vanishes at an appropriate rate as the size $n$ of the system grows. Our analysis requires separate treatment for the cases $b<s$ and $b=s$. In the case $b<s$ we prove that when the inverse temperature is taken to be of the form $β_{n}=\widehatβ (b/s)^{n/2}$ for $\widehatβ>0$, the normalized partition function of the system converges weakly as $n \to \infty$ to a distribution $\mathbf{L}(\widehatβ)$ depending continuously on the parameter $\widehatβ$. In the case $b=s$ we find a critical point in the behavior of the model when the inverse temperature is scaled as $β_{n}=\widehatβ/n$; for an explicitly computable critical value $κ_{b} > 0$ the variance of the normalized partition function converges to zero with large $n$ when $\widehatβ\leq κ_{b}$ and grows without bound when $\widehatβ>κ_{b}$. Finally, we prove a central limit theorem for the normalized partition function when $\widehatβ\leq κ_{b}$.

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