If W is the simple random walk on the square lattice Z2, then W induces a random walk WG on any spanning subgraph G ⊂ Z2 of the lattice as follows: viewing W as a uniformly random infinite word on the alphabet {x, −x, y, −y}, the walk WG starts at the origin and follows the directions specified by W, only accepting steps of W along which the walk WG does not exit G. For any fixed G ⊂ Z2, the walk WG is distributed as the simple random walk on G, and hence WG is almost surely recurrent in the sense that WG visits every site reachable from the origin in G infinitely often. This fact naturally leads us to ask the following: does W almost surely have the property that WG is recurrent for every G ⊂ Z2? We answer this question negatively, demonstrating that exceptional subgraphs exist almost surely. In fact, we show more to be true: exceptional subgraphs continue to exist almost surely for a countable collection of independent simple random walks, but on the other hand, there are almost surely no exceptional subgraphs for a branching random walk.@en

Two permutations σ and π are ℓ-similar if they can be decomposed into subpermutations σ(1), . . . ,σ(ℓ) and π(1), . . . ,π(ℓ) such that σ(i) is order-isomorphic to π(i) for all i ∈[ℓ]. Recently, Dudek, Grytczuk, and Ruciński Variations on twins in permutations, Electron. J. Combin., 28 (2021), P3.19. posed the problem of determining the minimum ℓ for which two permutations chosen independently and uniformly at random are ℓ-similar. We show that two such permutations are O(n1/3 log11/6(n))-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalizes to simultaneous decompositions of multiple permutations.@en