Capacity of the range of random walk on $\mathbb {Z}^d$
HTML articles powered by AMS MathViewer
- by Amine Asselah, Bruno Schapira and Perla Sousi PDF
- Trans. Amer. Math. Soc. 370 (2018), 7627-7645 Request permission
Abstract:
We study the capacity of the range of a transient simple random walk on $\mathbb {Z}^d$. Our main result is a central limit theorem for the capacity of the range for $d\ge 6$. We present a few open questions in lower dimensions.References
- Amine Asselah and Bruno Schapira, Boundary of the range of transient random walk, Probab. Theory Related Fields 168 (2017), no. 3-4, 691–719. MR 3663629, DOI 10.1007/s00440-016-0722-4
- Amine Asselah and Bruno Schapira, Moderate deviations for the range of a transient random walk: path concentration, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 3, 755–786 (English, with English and French summaries). MR 3665554, DOI 10.24033/asens.2331
- Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010. MR 2722836, DOI 10.1017/CBO9780511779398
- J. M. Hammersley, Generalization of the fundamental theorem on sub-additive functions, Proc. Cambridge Philos. Soc. 58 (1962), 235–238. MR 137800, DOI 10.1017/s030500410003646x
- N. Jain and S. Orey, On the range of random walk, Israel J. Math. 6 (1968), 373–380 (1969). MR 243623, DOI 10.1007/BF02771217
- Naresh C. Jain and S. Orey, Some properties of random walk paths, J. Math. Anal. Appl. 43 (1973), 795–815. MR 359004, DOI 10.1016/0022-247X(73)90293-X
- Gregory F. Lawler, Intersections of random walks, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1991. MR 1117680
- Gregory F. Lawler and Vlada Limic, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010. MR 2677157, DOI 10.1017/CBO9780511750854
- J.-F. Le Gall, Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection, Comm. Math. Phys. 104 (1986), no. 3, 471–507 (French, with English summary). MR 840748
- David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson. MR 2466937, DOI 10.1090/mbk/058
- Balázs Ráth and Artëm Sapozhnikov, Connectivity properties of random interlacement and intersection of random walks, ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), 67–83. MR 2889752
- Alain-Sol Sznitman, Vacant set of random interlacements and percolation, Ann. of Math. (2) 171 (2010), no. 3, 2039–2087. MR 2680403, DOI 10.4007/annals.2010.171.2039
Additional Information
- Amine Asselah
- Affiliation: Aix-Marseille Université – and – Université Paris-Est Créteil, Laboratoire d’Analyse et de Mathématiques Appliquées, Bât. P2, 94010 Créteil Cedex, France
- Email: amine.asselah@u-pec.fr
- Bruno Schapira
- Affiliation: Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France
- MR Author ID: 798213
- Email: bruno.schapira@univ-amu.fr
- Perla Sousi
- Affiliation: University of Cambridge, Cambridge, Department of Pure Mathematics, CB3 OWB, United Kingdom
- Email: p.sousi@statslab.cam.ac.uk
- Received by editor(s): February 10, 2016
- Received by editor(s) in revised form: January 12, 2017
- Published electronically: April 25, 2018
- Additional Notes: We thank the Institute IMéRA in Marseille for its hospitality.
This work has been carried out thanks partially to the support of A$^*$MIDEX grant (ANR-11-IDEX-0001-02) funded by the French Government “Investissements d’Avenir” program. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 7627-7645
- MSC (2010): Primary 60F05, 60G50
- DOI: https://doi.org/10.1090/tran/7247
- MathSciNet review: 3852443