リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree

Barlow, M. T. Croydon, D. A. Kumagai, T. 京都大学 DOI:10.1007/s00440-021-01078-w

2021.11

概要

This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonal part of the averaged heat kernel and the expected distance travelled by the associated simple random walk.

この論文で使われている画像

参考文献

[1] S. Athreya, W. L¨

ohr, and A. Winter, The gap between Gromov-vague and GromovHausdorff-vague topology, Stochastic Process. Appl. 126 (2016), no. 9, 2527–2553.

, Invariance principle for variable speed random walks on trees, Ann. Probab. 45

[2]

(2017), no. 2, 625–667.

[3] M. T. Barlow, Random walks and heat kernels on graphs, London Mathematical Society

Lecture Note Series, vol. 438, Cambridge University Press, Cambridge, 2017.

[4] M. T. Barlow, D. A. Croydon, and T. Kumagai, Subsequential scaling limits of simple

random walk on the two-dimensional uniform spanning tree, Ann. Probab. 45 (2017), no. 1,

4–55.

[5] M. T. Barlow and A. A. J´

arai, Geometry of uniform spanning forest components in high

dimensions, Canad. J. Math. (2019), 1–25.

45

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

[6] M. T. Barlow, A. A. J´

arai, T. Kumagai, and G. Slade, Random walk on the incipient infinite

cluster for oriented percolation in high dimensions, Comm. Math. Phys. 278 (2008), no. 2,

385–431.

[7] M. T. Barlow and T. Kumagai, Random walk on the incipient infinite cluster on trees,

Illinois J. Math. 50 (2006), no. 1-4, 33–65.

[8] M. T. Barlow and R. Masson, Exponential tail bounds for loop-erased random walk in two

dimensions, Ann. Probab. 38 (2010), no. 6, 2379–2417. MR 2683633

, Spectral dimension and random walks on the two dimensional uniform spanning

[9]

tree, Comm. Math. Phys. 305 (2011), no. 1, 23–57.

[10] M. T. Barlow and M. Murugan, Stability of the elliptic harnack inequality, Ann. Math. 187

(2018), 1–47.

[11] I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, Uniform spanning forests, Ann. Probab.

29 (2001), no. 1, 1–65.

[12] K. Burdzy and Z.-Q. Chen, Weak convergence of reflecting Brownian motions, Electron.

Comm. Probab. 3 (1998), 29–33.

[13] D. A. Croydon, Scaling limits of stochastic processes associated with resistance forms, Ann.

Inst. Henri Poincar´e Probab. Stat. 54 (2018), no. 4, 1939–1968.

[14] D. A. Croydon and B. M. Hambly, Local limit theorems for sequences of simple random

walks on graphs, Potential Anal. 29 (2008), no. 4, 351–389.

, Local limit theorems for sequences of simple random walks on graphs, Potential

[15]

Anal. 29 (2008), no. 4, 351–389.

[16] D. A. Croydon and T. Kumagai, Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive, Electron. J. Probab. 13 (2008), no. 51,

1419–1441.

[17] T. Duquesne and J.-F. Le Gall, Probabilistic and fractal aspects of L´evy trees, Probab.

Theory Related Fields 131 (2005), no. 4, 553–603.

[18] N. Holden and X. Sun, SLE as a mating of trees in Euclidean geometry, Comm. Math.

Phys. 364 (2018), no. 1, 171–201.

[19] R. Kenyon, The asymptotic determinant of the discrete Laplacian, Acta Math. 185 (2000),

no. 2, 239–286.

[20] J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer.

Math. Soc. 216 (2012), no. 1015, vi+132.

[21] T. Kumagai, Random walks on disordered media and their scaling limits, Lecture Notes in

Mathematics, vol. 2101, Springer, Cham, 2014, Lecture notes from the 40th Probability

´ e de Probabilit´es de Saint-Flour.

Summer School held in Saint-Flour, 2010, Ecole

d’Et´

[22] T. Kumagai and J. Misumi, Heat kernel estimates for strongly recurrent random walk on

random media, J. Theoret. Probab. 21 (2008), no. 4, 910–935.

[23] G. F. Lawler, A self-avoiding random walk, Duke Math. J. 47 (1980), no. 3, 655–693.

, Intersections of random walks, Probability and its Applications, Birkh¨auser Boston,

[24]

Inc., Boston, MA, 1991.

, Loop-erased random walk, Perplexing problems in probability, Progr. Probab.,

[25]

vol. 44, Birkh¨auser Boston, Boston, MA, 1999, pp. 197–217.

[26]

, The probability that planar loop-erased random walk uses a given edge, Electron.

Commun. Probab. 19 (2014), no. 51, 13.

[27] G. F. Lawler and V. Limic, The Beurling estimate for a class of random walks, Electron.

J. Probab. 9 (2004), no. 27, 846–861.

46

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.

G. F. Lawler, O. Schramm, and W. Werner, Conformal invariance of planar loop-erased

random walks and uniform spanning trees, Ann. Probab. 32 (2004), no. 1B, 939–995.

G. F. Lawler and F. Viklund, Convergence of loop-erased random walk in the natural

parametrization, preprint available at arXiv.org/1603.05203.

J.-F. Le Gall, Random real trees, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 1, 35–62.

R. Lyons and Y. Peres, Probability on trees and networks, Cambridge Series in Statistical

and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016.

R. Masson, The growth exponent for planar loop-erased random walk, Electron. J. Probab.

14 (2009), no. 36, 1012–1073.

R. Pemantle, Choosing a spanning tree for the integer lattice uniformly, Ann. Probab. 19

(1991), no. 4, 1559–1574.

O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel

J. Math. 118 (2000), 221–288.

D. B. Wilson, Generating random spanning trees more quickly than the cover time, Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), ACM, New York, 1996, pp. 296–303.

Martin T. Barlow

Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2,

Canada.

E-mail: barlow@math.ubc.ca

David A. Croydon

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.

E-mail: croydon@kurims.kyoto-u.ac.jp

Takashi Kumagai

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.

E-mail: kumagai@kurims.kyoto-u.ac.jp

47

...

参考文献をもっと見る