Two-Dimensional Random Interlacements and Late Points for Random Walks

Abstract

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.

Transience of conditioned walks on the plane: encounters and speed of escape

“…The conditioned walk S then became an interesting object on its own. Some of its (sometimes surprising) properties shown in [6,9] for any x 0 = 0;…”

“…The next proposition is taken from [9] and presents some of the basic properties of an S-walk that were proved in [6]. We also add an explicit expression for G(x, y) = E x [# visits to y], the Green function of an S-walk at item (vi).…”

Conditioned Two-Dimensional Simple Random Walk: Green’s Function and Harmonic Measure

“…Regarding item (ii), it is important to keep in mind that the harmonic measure hm A with respect to the conditioned walk is generally different from the harmonic measure hm A with respect to the simple random walk. The two-dimensional simple random walk conditioned on never hitting the origin is the main ingredient in the construction of the two-dimensional random interlacements introduced in [3] and further studied in [2,10] (by its turn, it is an extention of classical random intelacement model [1,4,11] to two dimensions). It then became evident that the conditioned walk (denoted by S in this paper) is an interesting object on its own.…”

“…, which indeed leads to (10). However, the proof of (7)- (8) in [3] is somewhat involved and not very intuitive. Here, we take the "classical" route of first obtaining the expression for the Green's function; then, it is straightforward to derive (7)- (8) from it in the usual way.…”

Two-Dimensional Brownian Random Interlacements

“…At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [12,13], as well as the results specific to the continuous case.…”