In this talk, I will present joint works with Francis Comets and Serguei Popov.
How can one condition the two-dimensional simple random walk (2D-SRW), which is well known to be recurrent, on not hitting the origin ?
This question will be answered by defining rigorously the 2D-SRW conditioned to avoid the origin. I will then discuss some of its surprising properties and recent results concerning its speed of escape towards infinity.
I will also present the random interlacements model build upon the conditioned 2D-SRW, which was introduced by the aforementioned coauthors (together with Marina Vachkovskaia) in 2015. I will discuss some properties of this model : FKG inequality, 0-1 law, phase transition and behavior at criticality.
During the talk, I will also evoke the continuous counterparts of these processes, namely : the two-dimensional Brownian motion conditioned to avoid the unit ball and the two-dimensional random interlacements. If time allows, I will mention some results on the Brownian random interlacements.