Scaling limits for random walks in random conductances

随机电导上的随机游动形成了一类已被深入研究的随机环境中的运动。在电导远离零和无穷大的情况下，众所周知，重新标度的游动收敛于布朗运动。在这种情况下，这个模型展示了“陷阱现象”，即由于环境中存在表现不典型的小区域而导致行走速度减慢。在这种情况下，通常的不变性原则不再适用。虽然这一机制现在已经被很好地识别，但在捕获区域中获得功能尺度极限和其他精细结果(例如老化、猝灭收敛)是具有挑战性的。卡洛的项目探索了随机环境和随机图中随机行走的陷阱现象。特别是，卡罗将研究Bouchaud陷阱模型和囚禁体系中随机电导上的随机游动之间的关系。研究工作将从尝试修改现有的一维随机电导中有偏随机游动的结果开始，遵循第一个结果的作者给出的建议(Q.Berger和M.Sali，2020)。这意味着证明，对于几乎每一次对环境的认识，以环境为条件的行走定律都收敛于某种(已知的)分布。第二个项目将涉及一维不偏不倚的行走。这里的研究假设是，标度因子和极限等于相应的无偏Bouchaud陷阱模型中出现的和由Fontes，Isopi和Newman在2002年引入的。这两个模型的分析因行走方向的存在或不存在而有很大不同。特别是，直观地说，偏向行走最终不会回溯太多，使每个陷阱的访问次数有限。这一事实在偏向行走的分析中被大量使用，而对于不偏向的情况显然不是真的，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上的，我们不期望进行任何数据分析，即使是对这些类型的模型的模拟也非常有用，最终可能只是为了解释目的而进行。

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.