标题： 分数Anderson模型的Green函数衰减及其与长程自避行走的联系 显示英文标题

标题： Decay of the Green's function of the fractional Anderson model and connection to long-range SAW

摘要： 我们证明了分数阶Anderson模型的Green函数与具有长程跳跃的自回避随机游走的两点函数之间的一个联系，采用了Schenker在2015年提出的策略。 这一联系使我们可以利用自回避随机游走理论的结果来改进之前对于强无序下分数阶Anderson模型已知的界。 特别是，我们扩展了谱局域化发生的无序参数范围。 此外，我们证明了在强无序下任意的$0<\alpha <1$的Green函数衰减可以任意接近相应分数阶Laplacian的有质量resolvent的衰减，这与标准Anderson模型$\alpha =1$的情况一致。 我们还得到了离散分数阶Laplacian的resolvent（任意质量为$m\geq 0,$）的上下界，这些界本身也具有独立的兴趣。 显示英文摘要

摘要： We prove a connection between the Green's function of the fractional Anderson model and the two point function of a self-avoiding random walk with long range jumps, adapting a strategy proposed by Schenker in 2015. This connection allows us to exploit results from the theory of self-avoiding random walks to improve previous bounds known for the fractional Anderson model at strong disorder. In particular, we enlarge the range of the disorder parameter where spectral localization occurs. Moreover we prove that the decay of Green's function at strong disorder for any $0<\alpha <1$ is arbitrarily close to the decay of the massive resolvent of the corresponding fractional Laplacian, in agreement with the case of the standard Anderson model $\alpha =1$. We also derive upper and lower bounds for the resolvent of the discrete fractional Laplacian with arbitrary mass $m\geq 0,$ that are of independent interest.