Scaling limits for random walks in random conductances

随机电导下的随机行走形成了随机环境中一类完善的运动，已经得到了深入的研究。在电导从零到无穷大有界的情况下，众所周知，重新缩放的行走收敛于布朗运动。在这种情况下，该模型展示了“捕获现象”，即由于环境中存在非典型行为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。虽然这种机制现在已经很好地确定了，但在被困状态下获得功能缩放极限和其他精细结果（例如老化，淬火收敛）是具有挑战性的。Carlo的项目探讨了随机环境和随机图中随机游走的陷阱现象。特别是，Carlo将研究Bouchaud陷阱模型与陷阱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率从退火到淬火的有偏随机游走的现有结果开始，遵循第一个结果的作者给出的建议（Q. Berger和M. Salvi, 2020）。这意味着证明，对于几乎每一个环境的实现，以环境为条件的行走法则收敛于某些（已知的）分布。第二个项目将涉及一维的无偏行走。在这里，研究假设是比例因子和极限等于相应的无偏Bouchaud's Trap模型中出现的比例因子和极限，这些比例因子和极限是由Fontes， Isopi和Newman在2002年引入的。这两种模型的分析结果在很大程度上取决于行走方向的存在与否。特别是，直观地说，有偏差的行走最终不会“太多”地回溯，从而使每个陷阱的访问次数有限。这一事实在有偏行走的分析中被大量使用，但对于无偏行走的情况显然不是这样，因此用于一种情况的技术不能无缝地转化为另一种情况。这项研究几乎完全是理论性的，我们不期望进行任何数据分析，甚至这些类型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。

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.