Extremal points of random walks on planar and tree graphs

Abstract: I will review recent progress on the description of points heavily visited by paths of random walks. The focus will be on the situations where the random walk has an approximate scale-invariant structure and the associated local time process is thus logarithmically correlated in space. Two geometric settings will be considered: the simple random walk on finite subsets of the square lattice and the random walk on homogeneous trees of finite depth. In the former case, a full description will be given of the scaling limit of thick, thin and avoided points for the walk run up to the times proportional to the cover time. For the latter setting, the law of the most frequently visited leaf-vertex, and the time spent there, will be given in the limit of the tree depth tending to infinity. In both cases, the spatial laws will be determined by a corresponding multiplicative chaos measure. Based on joint papers with Y. Abe, S. Lee and O. Louidor.

mathematical physics

Audience: researchers in the discipline

( video )

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