# zbMATH — the first resource for mathematics

Second-order term of cover time for planar simple random walk. (English) Zbl 07374950
Summary: We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length $$n$$. Dembo et al. (Ann Math 160:433–464, 2004) identified the leading term in the asymptotics for the cover time as $$n$$ goes to infinity. In this paper, we study the exact second order term. This is a discrete analogue of the work on the cover time for planar Brownian motion by Belius and Kistler (Probab Theory Relat Fields 167:461–552, 2017).
##### MSC:
 60G50 Sums of independent random variables; random walks 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J65 Brownian motion
Full Text:
##### References:
 [1] Belius, D.; Kistler, N., The subleading order of two dimensional cover times, Probab. Theory Relat. Fields, 167, 461-552 (2017) · Zbl 1365.60071 [2] Belius, D.; Rosen, J.; Zeitouni, O., Barrier estimates for a critical Galton-Watson process and the cover time of the binary tree, Ann. Inst. Henri Poincaré., 55, 127-154 (2019) · Zbl 1447.60133 [3] Belius, D.; Rosen, J.; Zeitouni, O., Tightness for the cover time of the two dimensional sphere, Probab. Theory Relat. Fields (2019) · Zbl 1434.60222 [4] Bramson, M.: Convergence of solutions of the Kolmogorov equation to traveling waves. Amer. Math. Soc. (1983). doi:10.1090/memo/0285 · Zbl 0517.60083 [5] Comets, F., Gallesco, C., Popov, S., Vachkovskaia, M.: On large deviations for the cover time of two-dimensional torus. Electron. J. Probab. (2013). doi:10.1214/EJP.v18-2856 · Zbl 1294.60066 [6] Comets, F.; Popov, S.; Vachkovskaia, M., Two-dimensional random interlacements and late points for random walks, Commun. Math. Phys., 343, 129-164 (2016) · Zbl 1336.60185 [7] Comets, F.; Popov, S., The vacant set of two-dimensional critical random interlacement is infinite, Ann. Probab., 45, 4752-4785 (2017) · Zbl 1409.60140 [8] Dembo, A.; Peres, Y.; Rosen, J.; Zeitouni, O., Cover times for Brownian motion and random walks in two dimensions, Ann. Math., 160, 433-464 (2004) · Zbl 1068.60018 [9] Dembo, A.; Peres, Y.; Rosen, J.; Zeitouni, O., Late points for random walks in two dimensions, Ann. Probab., 34, 219-263 (2006) · Zbl 1100.60057 [10] Ding, J.: On cover times for 2D lattices. Electron. J. Probab. (2012). doi:10.1214/EJP.v17-2089 · Zbl 1258.60044 [11] Ding, J.; Lee, JR; Peres, Y., Cover times, blanket times, and majorizing measures, Ann. Math., 175, 1409-1471 (2012) · Zbl 1250.05098 [12] Fitzsimmons, PJ; Pitman, J., Kac’s moment formula and the Feynman-Kac formula for additive functionals of a Markov process, Stoch. Process. Appl., 79, 117-134 (1999) · Zbl 0962.60067 [13] Lawler, G., Intersections of Random Walks (1991), Boston: Birkhäuser, Boston · Zbl 1228.60004 [14] Levin, DA; Peres, Y., Markov Chains and Mixing Times, Second Edition with Contributions by Elizabeth L. Wilmer with an Appendix Written by James G. Propp and David B. Wilson (2017), Providence: American Mathematical Society, Providence [15] Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics, 42. Cambridge University Press, New York (2016). http://pages.iu.edu/ rdlyons/ · Zbl 1376.05002 [16] Okada, I., Geometric structures of late points of a two-dimensional simple random walk, Ann. Probab., 47, 5, 2869-2893 (2019) · Zbl 1448.60028 [17] Rodriguez, P-F, On pinned fields, interlacements, and random walk on $$({\mathbb{Z}}/N{\mathbb{Z}})^2$$, Probab. Theory Relat. Fields, 173, 3-4, 1265-1299 (2019) · Zbl 1411.60040 [18] Ueno, T., On recurrent Markov processes, Kōdai Math. Sem. Rep., 12, 109-142 (1960) · Zbl 0094.32201 [19] Zhai, A.: Exponential concentration of cover times. Electron. J. Probab. (2018). doi:10.1214/18-EJP149 · Zbl 1391.60177
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.