The intermediate disorder regime for a directed polymer model on a hierarchical lattice
The intermediate disorder regime for a directed polymer model on a hierarchical lattice
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DOI：
10.1016/j.spa.2017.02.011
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发表时间：
2015-08
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通讯作者：
Tom Alberts;J. Clark;S. Kocić
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文献类型：
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作者：
Tom Alberts;J. Clark;S. Kocić
本文研究了一个定义在分级菱形格上的定向聚合物模型，该格是通过依赖于分支数B∈ N和链段数s∈ N的配方递归构造的.当B≤ s时，已知该模型对所有的反温度β正值都表现出强无序，因此弱无序只在β= 0（无限温度）时占主导地位。我们的重点是所谓的中间无序区，其中逆温度β <$β n随着系统尺寸n的增长以适当的速率消失。我们的分析需要对B< s和B= s的情况分别处理。在B< s的情况下，我们证明了当逆温度取β n= β（B/s）n/2（β> 0）时，系统的归一化配分函数在n→∞时弱收敛到一个分布L（β），并且对于初始权分布是普适的。我们证明的收敛性使用重整化群类型的想法，而不是标准的维纳混沌分析。在B= s的情况下，我们发现当逆温度被标度为β n= β/n时，模型的行为中存在一个临界点;对于一个显式可计算的临界值κ B> 0，当β ≤ κ B时，归一化配分函数的方差随着n的增大而收敛到零，当β> κ B时，归一化配分函数的方差则无限增长。最后证明了当β B ≤ κ时，正规化配分函数的中心极限定理.
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∈ N and a segment number s∈ N. When b≤ s it is known 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≡ 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= β ̂ (b/s) n/2 for β ̂> 0, the normalized partition function of the system converges weakly as n→∞ to a distribution L (β ̂) and does so universally with respect to the initial weight distribution. We prove the convergence using renormalization group type ideas rather than the standard Wiener chaos analysis. 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= β ̂/n; for an explicitly computable critical value κ b> 0 the variance of the normalized partition function converges to zero with large n when β ̂≤ κ b and grows without bound when β ̂> κ b. Finally, we prove a central limit theorem for the normalized partition function when β ̂≤ κ b.