×

zbMATH — the first resource for mathematics

The subleading order of two dimensional cover times. (English) Zbl 1365.60071
The \(\varepsilon \)-cover time of the two-dimensional torus by Brownian motion is the time it takes for the process to come within distance \(\varepsilon > 0\) from any point. A. Dembo et al. [Ann. Math. (2) 160, No. 2, 433–464 (2004; Zbl 1068.60018)] proved the law of large numbers: \(\frac{{{C_\varepsilon }}}{{\frac{1}{\pi }\log {\varepsilon ^{ - 1}}}} = (1 + o(1))\log{\varepsilon ^{ - 2}}\), \(\varepsilon \to 0\). The authors improve this result: \(\frac{{{C_\varepsilon }}}{{\frac{1}{\pi }\log {\varepsilon ^{ - 1}}}} = \log {\varepsilon ^{ - 2}} - (1 + o(1))\log\log{\varepsilon ^{ - 1}}\) in probability, as \(\varepsilon \to 0\).

MSC:
60J65 Brownian motion
60G70 Extreme value theory; extremal stochastic processes
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G50 Sums of independent random variables; random walks
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Aïdékon, E; Berestycki, J; Brunet, É; Shi, Z, Branching Brownian motion seen from its tip, Probab. Theory Related Fields, 157, 405-451, (2013) · Zbl 1284.60154
[2] Arguin, L-P; Bovier, A; Kistler, N, Genealogy of extremal particles of branching Brownian motion, Commun. Pure Appl. Math., 64, 1647-1676, (2011) · Zbl 1236.60081
[3] Arguin, L-P; Bovier, A; Kistler, N, The extremal process of branching Brownian motion, Probab. Theory Related Fields, 157, 535-574, (2013) · Zbl 1286.60045
[4] Belius, D, Gumbel fluctuations for cover times in the discrete torus, Probab. Theory Related Fields, 157, 635-689, (2013) · Zbl 1295.60053
[5] Bramson, M, Maximal displacement of branching Brownian motion, Commun. Pure Appl. Math, 31, 531-581, (1978) · Zbl 0361.60052
[6] Bramson, M, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Am. Math. Soc., 44, 1-190, (1983) · Zbl 0517.60083
[7] Bramson, M; Zeitouni, O, Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field, Commun. Pure Appl. Math., 65, 1-20, (2012) · Zbl 1237.60041
[8] Carr, P; Schröder, M, Bessel processes, the integral of geometric Brownian motion, and Asian options, Teor. Veroyatnost. i Primenen., 48, 503-533, (2003) · Zbl 1056.91026
[9] Comets, F., Gallesco, C., Popov, S., Vachkovskaia, M.: On large deviations for the cover time of two-dimensional torus. Electron. J. Probab. 18(96), 18 (2013) · Zbl 1294.60066
[10] Dembo, A; Peres, Y; Rosen, J, Brownian motion on compact manifolds: cover time and late points, Electron. J. Probab., 8, 1-14, (2003) · Zbl 1063.58021
[11] Dembo, A., Yuval, Peres, Y., Rosen, J., Zeitouni, O.: Cover times for Brownian motion and random walks in two dimensions. Ann. Math. 160(2), 433-464 (2004) · Zbl 1068.60018
[12] 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
[13] Ding, J, On cover times for 2D lattices, Electron. J. Probab., 17, 18, (2012) · Zbl 1258.60044
[14] Ding, J, Asymptotics of cover times via Gaussian free fields: bounded-degree graphs and general trees, Ann. Probab., 42, 464-496, (2014) · Zbl 1316.60064
[15] Ding, J; Zeitouni, O, A sharp estimate for cover times on binary trees, Stochastic Process. Appl., 122, 2117-2133, (2012) · Zbl 1255.05179
[16] Eisenbaum, N; Kaspi, H; Marcus, MB; Rosen, J; Shi, Z, A ray-knight theorem for symmetric Markov processes, Ann. Probab., 28, 1781-1796, (2000) · Zbl 1044.60064
[17] Fitzsimmons, PJ; Pitman, J, Kac’s moment formula and the Feynman-Kac formula for additive functionals of a Markov process, Stochastic Process. Appl., 79, 117-134, (1999) · Zbl 0962.60067
[18] Goodman, J; den Hollander, F, Extremal geometry of a Brownian porous medium, Probab. Theory Relat Fields, 160, 127-174, (2013) · Zbl 1327.60037
[19] Kistler, N.: Derrida’s random energy models. Lecture Notes in Mathematics, vol. 2143. Springer, Berlin (2015) · Zbl 1338.60231
[20] Lawler, G.F.: Intersections of Random Walks. Probability and its Applications. Birkhäuser Boston Inc., Boston (1991) · Zbl 1228.60004
[21] Lawler, G.F.: On the covering time of a disc by simple random walk in two dimensions. In: Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), Progr. Probab., vol. 33, pp. 189-207. Birkhäuser Boston (1993) · Zbl 0789.60019
[22] Marcus, M.B., Rosen, J.: Markov processes, Gaussian processes, and local times, Cambridge Studies in Advanced Mathematics, vol. 100. Cambridge University Press, Cambridge (2006) · Zbl 1129.60002
[23] Matthews, P, Covering problems for Brownian motion on spheres, Ann. Probab., 16, 189-199, (1988) · Zbl 0638.60014
[24] Mörters, P., Peres, Y.: Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010) (with an appendix by Oded Schramm and Wendelin Werner) · Zbl 1056.91026
[25] Pitman, J; Yor, M, A decomposition of Bessel bridges, Z. Wahrsch. Verw. Gebiete, 59, 425-457, (1982) · Zbl 0484.60062
[26] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, vol. 293, 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006
[27] Scheike, TH, A boundary-crossing result for Brownian motion, J. Appl. Probab., 29, 448-453, (1992) · Zbl 0806.60065
[28] Sznitman, A.-S.: Topics in occupation times and Gaussian free fields. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2012) · Zbl 1246.60003
[29] Ueno, T, On recurrent Markov processes, Kōdai Math. Sem. Rep., 12, 109-142, (1960) · Zbl 0094.32201
[30] Webb, C, Exact asymptotics of the freezing transition of a logarithmically correlated random energy model, J. Stat. Phys., 145, 1595-1619, (2011) · Zbl 1231.82091
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.