Scaling limits for random walks in random conductances

随机电导中随机游走的缩放限制

• 批准号: 2284054
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2284054
• 关键词: Scaling; limits; random; walks; conductances

Random walks on random conductances form a well-established class of motions in random environment that has already been investigated in depth. In the case where the conductances are bounded away from zero and infinity, it is well-known that the re-scaled walk converges to a Brownian motion. Out of this regime this model showcases "trapping phenomena", that is, a slow down in the walk due to the presence of small areas in the environment that behave atypically. In this case, the usual invariance principle does not hold anymore. Although this mechanism is now well identified, it is challenging to obtain, in the trapped regimes, functional scaling limits and other fine results (e.g. aging, quenched convergence). Carlo's project explores trapping phenomena for random walks in random environments and on random graphs. In particular, Carlo will investigate the relationship between the Bouchaud's Trap Model and random walks on random conductances in the trapped regimes.The research work will start with an attempt to modify the existing result for the biased random walk in random conductances in dimension one from annealed to quenched, following the footsteps of the suggestion given by the authors of the first result (Q. Berger and M. Salvi, 2020). This means proving that, for almost every realisation of the environment, the law of the walk, conditional on the environment, converges to some (known) distribution. The second project will involve the unbiased walk in dimension one. Here, the research hypothesis is that the scaling factor and limit are equal to the ones that appear in the corresponding unbiased Bouchaud's Trap Model and that have been introduced by Fontes, Isopi and Newman in 2002. The analyses of these two models differ substantially depending on the presence or absence of a direction for the walk. In particular, intuitively, the biased walk does not backtrack "too much" eventually, making the number of visits to each trap finite. This fact is heavily used in the analysis of the biased walk and it is evidently not true for the unbiased case, therefore the techniques used for one do not translate to the other seamlessly. This research is almost entirely theoretic and we do not expect to perform any data analysis, even simulations of these types of model have very limited usefulness and could eventually be performed just for explanatory purposes.

随机电导上的随机游动形成了随机环境中的一类已被深入研究的运动。在电导远离零和无穷大的情况下，众所周知，重新缩放的行走收敛到布朗运动。在这种状态下，该模型展示了“捕获现象”，即由于环境中存在小区域而导致行走减慢。在这种情况下，通常的不变性原理不再成立。虽然这种机制现在已经被很好地识别，但在捕获机制中获得功能缩放限制和其他精细结果（例如老化，淬火收敛）是具有挑战性的。Carlo的项目探索了随机环境和随机图中随机游动的捕获现象。特别地，Carlo将研究布绍陷阱模型和陷阱区中随机电导的随机游走之间的关系。研究工作将从尝试修改一维随机电导中有偏随机游走的现有结果开始，从退火到淬火，遵循第一个结果（Q. Berger和M. Salvi，2020）。这意味着证明，对于几乎每一个环境的实现，行走定律，以环境为条件，收敛于某个（已知的）分布。第二个项目将涉及一维中的无偏行走。在这里，研究假设是，比例因子和限制是相等的，出现在相应的无偏Bouchaud的陷阱模型，并已介绍了Fontes，Isopi和纽曼在2002年。这两个模型的分析有很大的不同，这取决于步行方向的存在或不存在。特别是，直觉上，有偏行走最终不会回溯“太多”，使得访问每个陷阱的次数有限。这一事实在有偏行走的分析中被大量使用，显然对于无偏情况并不正确，因此用于一种情况的技术不能无缝地转换到另一种情况。这项研究几乎完全是理论性的，我们不希望进行任何数据分析，即使是这些类型的模型的模拟也非常有限，最终可能只是为了解释目的而进行。