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### Abstract

We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is “almost recurrent” in the sense that each infinite set is visited infinitely often, almost surely. We prove that, for a “large” set, the proportion of its sites visited by the conditioned walk is approximately a Uniform$[0,1]$ random variable. Also, given a set $G\subset {\mathbb{R}}^{2}$ that does not “surround” the origin, we prove that a.s. there is an infinite number of $k$’s such that $kG\cap {\mathbb{Z}}^{2}$ is unvisited. These results suggest that the range of the conditioned walk has “fractal” behavior.

### References

[CP17] The vacant set of two-dimensional critical random interlacement is infinite, Ann. Probab., Volume 45 (2017) no. 6B, pp. 4752-4785 | Zbl 1409.60140

[CPV16] Two-dimensional random interlacements and late points for random walks, Commun. Math. Phys., Volume 343 (2016) no. 1, pp. 129-164 | Zbl 1336.60185

[DRS14] An introduction to random interlacements, Springer, SpringerBriefs in Mathematics (2014) | Zbl 1304.60008

[KS64] A note on the Borel-Cantelli lemma, Ill. J. Math., Volume 8 (1964), pp. 248-251 | Zbl 0139.35401

[LL10] Random walk: a modern introduction, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 123 (2010) | Zbl 1210.60002

[MPW17] Non-homogeneous random walks: Lyapunov function methods for near-critical stochastic systems, Cambridge University Press, Cambridge Tracts in Mathematics, Volume 209 (2017) | Zbl 1376.60005

[PT15] Soft local times and decoupling of random interlacements, J. Eur. Math. Soc., Volume 17 (2015) no. 10, pp. 2545-2593 | Zbl 1329.60342

[Rev84] Markov chains, North-Holland, North-Holland Mathematical Library, Volume 11 (1984) | Zbl 0539.60073

[Szn10] Vacant set of random interlacements and percolation, Ann. Math., Volume 171 (2010) no. 3, pp. 2039-2087 | Zbl 1202.60160

[Woe09] Denumerable Markov chains. Generating functions, boundary theory, random walks on trees, European Mathematical Society, EMS Textbooks in Mathematics (2009) | Zbl 1219.60001

[ČT12] From random walk trajectories to random interlacements, Sociedade Brasileira de Matemática, Ensaios Matemáticos, Volume 23 (2012) | Zbl 1269.60002