Continuum models of directed polymers on disordered diamond fractals in the critical case
Continuum models of directed polymers on disordered diamond fractals in the critical case
复制标题
临界情况下无序金刚石分形上定向聚合物的连续体模型
DOI：
10.1214/22-aap1783
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
J. Clark
中科院分区：
文献类型：
--
作者：
J. Clark
我们构造和研究了一系列连续统随机聚合物测度$\mathbf{M}_{r}$，对应于最近在具有边际相关无序的层次图上聚合物模型的弱耦合机制中导出的极限配分函数定律。连续统聚合物通过单位间隔$[0,1]$的等距嵌入识别为具有Hausdorff维数2的紧凑菱形分形，并且存在一个自然概率度量$\mu$，可识别为连续统聚合物空间中的“均匀”，$\Gamma$。当钻石分形的维数小于2时，随机测度$\mathbf{M}_{r}$的实现与亚临界测度相比，表现出很强的局域化特性。然而，根据纯粹测度$\mu$独立选择的两条有向路径$p,q\in \Gamma$只有有限多个概率为1的交叉点，无序积测度$ \mathbf{M}_{r}\times \mathbf{M}_{r}$ as的实现为路径对集$(p,q)$赋予了正权重，这些路径对集的交集集是不可数的，但Hausdorff维数为零。我们使用广义（对数）豪斯多夫测度对这些维零集的大小给出了更精细的表征。随机测量定律$\mathbf{M}_{r}$不能被构造为亚临界高斯乘法混沌，因为高斯场的耦合强度在形式意义上必须是无限的。
We construct and study a family of continuum random polymer measures $\mathbf{M}_{r}$ corresponding to limiting partition function laws recently derived in a weak-coupling regime of polymer models on hierarchical graphs with marginally relevant disorder. The continuum polymers are identified with isometric embeddings of the unit interval $[0,1]$ into a compact diamond fractal with Hausdorff dimension two, and there is a natural probability measure, $\mu$, identifiable as being `uniform' over the space of continuum polymers, $\Gamma$. Realizations of the random measures $\mathbf{M}_{r}$ exhibit strong localization properties in comparison to their subcritical counterparts when the diamond fractal has dimension less than two. Whereas two directed paths $p,q\in \Gamma$ chosen independently according to the pure measure $\mu$ have only finitely many intersections with probability one, a realization of the disordered product measure $ \mathbf{M}_{r}\times \mathbf{M}_{r}$ a.s. assigns positive weight to the set of pairs of paths $(p,q)$ whose intersection sets are uncountable but with Hausdorff dimension zero. We give a more refined characterization of the size of these dimension zero sets using generalized (logarithmic) Hausdorff measures. The law of the random measure $\mathbf{M}_{r}$ cannot be constructed as a subcritical Gaussian multiplicative chaos because the coupling strength to the Gaussian field would, in a formal sense, have to be infinite.
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影响因子：
2.6
作者：
F. Caravenna;Rongfeng Sun;Nikos Zygouras
通讯作者：
F. Caravenna;Rongfeng Sun;Nikos Zygouras
影响因子：
1.4
作者：
Patricia Alonso Ruiz
通讯作者：
Patricia Alonso Ruiz
关于$$(2 1)$$-维定向聚合物的矩和临界窗口中的随机热方程
影响因子：
2.4
作者：
Caravenna F
通讯作者：
Caravenna F
二维 SHE 的关键时刻
DOI：
10.2140/pmp.2021.2.179
发表时间：
2021
期刊：
Probability and Mathematical Physics
影响因子：
--
作者：
Gu, Yu;Quastel, Jeremy;Tsai, Li-Cheng
通讯作者：
Tsai, Li-Cheng