We give a new proof of a result of Rick Kenyon that the probability that an edge in the middle of an n × n square is used in a loop-erased walk connecting opposites sides is of order n−3/4. We, in fact, improve the result by showing that this estimate is correct up to multiplicative constants.

The loop-erased random walk (LERW) in Zd, d ≥ 2, dimensions is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the “middle” of the path.

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density ρ ∈ (0,∞). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p◦ when it is on a vacant site and probability p• when it is on an oc...

This paper compiles basic components of the construction of random groups and of the proof of their properties announced in [G10]. Justification of each step, as well as the interrelation between them, is straightforward by available techniques specific to each step. On the other hand, there are several ingredients that cannot be truly appreciated without extending the present framework. We sha...

In recent years quantum random walks have garnered much interest among quantum information researchers. Part of the reason is the prospect that many hard problems can be solved efficiently by employing algorithms based on quantum random walks, in the same way that classical random walks have played a central role in many hugely successful randomized algorithms. In this paper we introduce a new ...

Joël De Coninck, François Dunlop, Thierry Huillet Abstract: We consider random walks Xn in Z+, obeying a detailed balance condition, with a weak drift towards the origin when Xn ր ∞. We reconsider the equivalence in law between a random walk bridge and a 1+1 dimensional Solid-On-Solid bridge with a corresponding Hamiltonian. Phase diagrams are discussed in terms of recurrence versus wetting. A ...