School of Science Department of Mathematics 54 Random Walks and Percolation on Graphs Supervisor: NITZSCHNER Maximilian Alexander / MATH Student: LE Minh Thang / DSCT Course: UROP 1100, Summer The elephant random walk is a time-inhomogeneous model of random motion introduced by Schütz and Trimper (Phys. Rev. E 70, 045101, 2004) which has attracted significant attention due to its unusual properties compared to the simple random walk. Standard approaches to understanding its behavior typically do not apply to this model. In this report, we briefly provide an introduction to martingale theory and highlight its use to obtain some fundamental characteristics of the elephant random walk. In the last part of the report, we aim to explore the leading-order behavior of the cover time of large tori by the elephant random walk by employing a coupling method. Random Walks and Percolation on Graphs Supervisor: NITZSCHNER Maximilian Alexander / MATH Student: NG Kai Fung / MATH-SF Course: UROP 1100, Fall UROP 2100, Spring In this report, we provide a proof of the finite collision property of two independent random walks on a certain wedge comb with random teeth. The latter is a subgraph of ℤ2 obtained by attaching to the vertices of ℤ finite segments of a length governed by a given profile function. In particular, we extend the proof presented in Section 4 of Barlow, Peres, and Sousi (Ann. Inst. H. Poincaré Probab. Statist. 48(4):922–946, 2012), to show that if α > 1, then a wedge comb with profile {| | : ∈ℤ}, whose teeth are randomly deleted with a fixed probability and in an independent fashion, has the finite collision property almost surely. Random Walks and Percolation on Graphs Supervisor: NITZSCHNER Maximilian Alexander / MATH Student: ZHANG Baining / MATH-IRE Course: UROP 1100, Summer We study the quenched critical point ℎ�β, ( ) of the self-avoiding walk among random conductances, generalizing the results of Chino and Sakai (J. Stat. Phys. 163:754–764, 2016) to the case of infinite connected graphs of bounded degrees with i.i.d. conductances. We show that the connective constant and the quenched critical point are independent of the choice of the reference point. We also show that the quenched critical point is non-random and satisfies the inequality ℎ0 − [ ] ≤ℎ�β, ( ) almost surely.
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