The loop-erased random walk (LERW) in three 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.

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.

We show that the lamplighter group L has a system of generators for which spectrum discrete Laplacian on Cayley graph is union an interval and countable set isolated points accumulating to point outside this interval. This first example with infinitely many gaps in its graph. The result obtained by careful study spectral properties one-parametric family convolution operators L. Our results pure...

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...