论文标题
定向聚合物，进行非常重的随机步行
Directed Polymer for very heavy tailed random walks
论文作者
论文摘要
在目前的工作中，我们调查了随机环境（DPRE）中的定向聚合物的情况，当随机步行的增量用等于零的尾巴重尾（$ \ mathbf {p} [| x_1 | \ geq n] $比任何功率较慢）。该案例尚未在定向聚合物的背景下进行研究，并与简单的对称随机行走案例以及增量属于$α$稳定定律的吸引域的范围的案例呈现关键差异，其中$α\ in（0，2] $。我们在每个非常强大的障碍中都无法分配一个非常强大的能力，即在每个范围内均分配了一个非常强大的份额 - 我们的分配范围是一个分配的部分 - 我们的范围均可在范围内分配 - 我们的范围均可划分 - 我们的范围是零件。在低温下会融合到零）或不取决于随机步行的较高特性：我们建立了从弱疾病到弱混乱的相位不匹配的必要条件。
In the present work, we investigate the case of Directed Polymer in a Random Environment (DPRE), when the increments of the random walk are heavy-tailed with tail-exponent equal to zero ($\mathbf{P}[|X_1|\geq n]$ decays slower than any power of $n$). This case has not yet been studied in the context of directed polymers and present key differences with the simple symmetric random walk case and the cases where the increments belong to the domain of attraction of an $α$-stable law, where $α\in (0, 2]$. We establish the absence of a very strong disorder regime - that is, the free energy equals zero at every temperature - for every disorder distribution. We also prove that a strong disorder regime (partition function converging to zero at low temperature) may exist or not depending on finer properties of the random walk: we establish non-matching necessary and sufficient conditions for having a phase transition from weak to strong disorder. In particular our results imply that for this directed polymer model, very strong disorder is not equivalent to strong disorder, shedding a new light on a long standing conjecture concerning the original nearest-neighbor DPRE.
