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On the range of a two-dimensional conditioned simple random walk. (Sur l’amplitude d’une marche aléatoire simple bidimensionnelle conditionnée.) (English) Zbl 1454.60110

Summary: 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.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G50 Sums of independent random variables; random walks
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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References:

[1] Comets, Francis; Popov, Serguei, The vacant set of two-dimensional critical random interlacement is infinite, Ann. Probab., 45, 6, 4752-4785 (2017) · Zbl 1409.60140
[2] Comets, Francis; Popov, Serguei; Vachkovskaia, Marina, Two-dimensional random interlacements and late points for random walks, Commun. Math. Phys., 343, 1, 129-164 (2016) · Zbl 1336.60185
[3] Drewitz, Alexander; Ráth, Balázs; Sapozhnikov, Artëm, An introduction to random interlacements (2014), Springer · Zbl 1304.60008
[4] Kochen, Simon; Stone, Charles, A note on the Borel-Cantelli lemma, Ill. J. Math., 8, 248-251 (1964) · Zbl 0139.35401
[5] Lawler, Gregory F.; Limic, Vlada, Random walk: a modern introduction, 123 (2010), Cambridge University Press · Zbl 1210.60002
[6] Menshikov, Mikhail; Popov, Serguei; Wade, Andrew, Non-homogeneous random walks: Lyapunov function methods for near-critical stochastic systems, 209 (2017), Cambridge University Press · Zbl 1376.60005
[7] Popov, Serguei; Teixeira, Augusto, Soft local times and decoupling of random interlacements, J. Eur. Math. Soc., 17, 10, 2545-2593 (2015) · Zbl 1329.60342
[8] Revuz, Daniel, Markov chains, 11 (1984), North-Holland · Zbl 0539.60073
[9] Sznitman, Alain-Sol, Vacant set of random interlacements and percolation, Ann. Math., 171, 3, 2039-2087 (2010) · Zbl 1202.60160
[10] Woess, Wolfgang, Denumerable Markov chains. Generating functions, boundary theory, random walks on trees (2009), European Mathematical Society · Zbl 1219.60001
[11] Černý, Jiří; Teixeira, Augusto Q., From random walk trajectories to random interlacements, 23 (2012), Sociedade Brasileira de Matemática · Zbl 1269.60002
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