On the range of a two-dimensional conditioned simple random walk
Annales Henri Lebesgue, Volume 2 (2019), pp. 349-368.


Keywords random interlacements, range, transience, simple random walk, Doob’s $h$-transform


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 2 that does not “surround” the origin, we prove that a.s. there is an infinite number of k’s such that kG 2 is unvisited. These results suggest that the range of the conditioned walk has “fractal” behavior.


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

[CPV16] Comets, Francis; Popov, Serguei; Vachkovskaia, Marina Two-dimensional random interlacements and late points for random walks, Commun. Math. Phys., Volume 343 (2016) no. 1, pp. 129-164 | DOI | MR | Zbl

[DRS14] Drewitz, Alexander; Ráth, Balázs; Sapozhnikov, Artëm An introduction to random interlacements, SpringerBriefs in Mathematics, Springer, 2014 | Zbl

[KS64] Kochen, Simon; Stone, Charles A note on the Borel-Cantelli lemma, Ill. J. Math., Volume 8 (1964), pp. 248-251 | DOI | MR | Zbl

[LL10] Lawler, Gregory F.; Limic, Vlada Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, 123, Cambridge University Press, 2010 | MR | Zbl

[MPW17] Menshikov, Mikhail; Popov, Serguei; Wade, Andrew Non-homogeneous random walks: Lyapunov function methods for near-critical stochastic systems, Cambridge Tracts in Mathematics, 209, Cambridge University Press, 2017 | Zbl

[PT15] Popov, Serguei; Teixeira, Augusto Soft local times and decoupling of random interlacements, J. Eur. Math. Soc., Volume 17 (2015) no. 10, pp. 2545-2593 | DOI | MR | Zbl

[Rev84] Revuz, Daniel Markov chains, North-Holland Mathematical Library, 11, North-Holland, 1984 | Zbl

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

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

[ČT12] Černý, Jiří; Teixeira, Augusto Q. From random walk trajectories to random interlacements, Ensaios Matemáticos, 23, Sociedade Brasileira de Matemática, 2012 | MR | Zbl