Scaling limits of the three-dimensional uniform spanning tree and associated random walk. (English) Zbl 1486.60019
The topic of the article is the three-dimensional uniform spanning tree (UST). One of the main results states that the law of UST is tight with respect to the local Gromov-Hausdorff-Prohorov topology as well as with respect to the path ensemble topology. The authors also prove that there exists a certain subsequential scaling limit for this law. Some properties of any possible scaling limit are also established.
The authors also study the random walk on the three-dimensional UST, deriving its walk dimension and its spectral dimension. The tightness of its annealed law under rescaling is obtained.
For related two-dimensional results see [O. Schramm, Isr. J. Math. 118, 221–288 (2000; Zbl 0968.60093); M.T. Barlow et. al., Ann. Probab. 45, No. 1, 4–55 (2017; Zbl 1377.60022)].
The authors also study the random walk on the three-dimensional UST, deriving its walk dimension and its spectral dimension. The tightness of its annealed law under rescaling is obtained.
For related two-dimensional results see [O. Schramm, Isr. J. Math. 118, 221–288 (2000; Zbl 0968.60093); M.T. Barlow et. al., Ann. Probab. 45, No. 1, 4–55 (2017; Zbl 1377.60022)].
Reviewer: Ivan Podvigin (Novosibirsk)
MSC:
| 60D05 | Geometric probability and stochastic geometry |
| 60G50 | Sums of independent random variables; random walks |
| 60G57 | Random measures |
| 60K37 | Processes in random environments |
References:
| [1] | Abraham, R., Delmas, J.-F. and Hoscheit, P. (2013). A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18 14. · Zbl 1285.60004 · doi:10.1214/EJP.v18-2116 |
| [2] | Athreya, S., Eckhoff, M. and Winter, A. (2013). Brownian motion on \[\mathbb{R} \]-trees. Trans. Amer. Math. Soc. 365 3115-3150. · Zbl 1278.60009 · doi:10.1090/S0002-9947-2012-05752-7 |
| [3] | Athreya, S., Löhr, W. and Winter, A. (2017). Invariance principle for variable speed random walks on trees. Ann. Probab. 45 625-667. · Zbl 1388.60120 · doi:10.1214/15-AOP1071 |
| [4] | Barlow, M. T. (2017). Random Walks and Heat Kernels on Graphs. London Mathematical Society Lecture Note Series 438. Cambridge Univ. Press, Cambridge. · Zbl 1365.05002 · doi:10.1017/9781107415690 |
| [5] | Barlow, M. T., Coulhon, T. and Kumagai, T. (2005). Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Comm. Pure Appl. Math. 58 1642-1677. · Zbl 1083.60060 · doi:10.1002/cpa.20091 |
| [6] | Barlow, M. T., Croydon, D. A. and Kumagai, T. (2021). Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree. Preprint. Available at arXiv:2104.03462v1. · Zbl 1495.60095 |
| [7] | Barlow, M. T., Croydon, D. A. and Kumagai, T. (2017). Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree. Ann. Probab. 45 4-55. · Zbl 1377.60022 · doi:10.1214/15-AOP1030 |
| [8] | Barlow, M. T., Járai, A. A., Kumagai, T. and Slade, G. (2008). Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Comm. Math. Phys. 278 385-431. · Zbl 1144.82030 · doi:10.1007/s00220-007-0410-4 |
| [9] | Barlow, M. T. and Masson, R. (2011). Spectral dimension and random walks on the two dimensional uniform spanning tree. Comm. Math. Phys. 305 23-57. · Zbl 1223.05285 · doi:10.1007/s00220-011-1251-8 |
| [10] | Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York. · Zbl 0944.60003 · doi:10.1002/9780470316962 |
| [11] | Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI. · Zbl 0981.51016 · doi:10.1090/gsm/033 |
| [12] | Croydon, D. (2008). Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. Henri Poincaré Probab. Stat. 44 987-1019. · Zbl 1187.60083 · doi:10.1214/07-AIHP153 |
| [13] | Croydon, D. A. (2007). Heat kernel fluctuations for a resistance form with non-uniform volume growth. Proc. Lond. Math. Soc. (3) 94 672-694. · Zbl 1116.58025 · doi:10.1112/plms/pdl025 |
| [14] | Croydon, D. A. (2008). Volume growth and heat kernel estimates for the continuum random tree. Probab. Theory Related Fields 140 207-238. · Zbl 1133.62066 · doi:10.1007/s00440-007-0063-4 |
| [15] | Croydon, D. A. (2009). Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. Ann. Probab. 37 946-978. · Zbl 1219.60052 · doi:10.1214/08-AOP425 |
| [16] | Croydon, D. A. (2010). Scaling limits for simple random walks on random ordered graph trees. Adv. in Appl. Probab. 42 528-558. · Zbl 1202.60162 · doi:10.1239/aap/1275055241 |
| [17] | Croydon, D. A. (2018). Scaling limits of stochastic processes associated with resistance forms. Ann. Inst. Henri Poincaré Probab. Stat. 54 1939-1968. · Zbl 1417.60067 · doi:10.1214/17-AIHP861 |
| [18] | Depperschmidt, A., Greven, A. and Pfaffelhuber, P. (2011). Marked metric measure spaces. Electron. Commun. Probab. 16 174-188. · Zbl 1225.60009 · doi:10.1214/ECP.v16-1615 |
| [19] | Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monographs 22. Math. Assoc. America, Washington, DC. |
| [20] | Edgar, G. A. (1998). Integral, Probability, and Fractal Measures. Springer, New York. · Zbl 0893.28001 · doi:10.1007/978-1-4757-2958-0 |
| [21] | Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81-126. · Zbl 1086.60050 · doi:10.1007/s00440-004-0411-6 |
| [22] | Greven, A., Pfaffelhuber, P. and Winter, A. (2009). Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees). Probab. Theory Related Fields 145 285-322. · Zbl 1215.05161 · doi:10.1007/s00440-008-0169-3 |
| [23] | Holden, N. and Sun, X. (2018). SLE as a mating of trees in Euclidean geometry. Comm. Math. Phys. 364 171-201. · Zbl 1408.60073 · doi:10.1007/s00220-018-3149-1 |
| [24] | Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). Springer, New York. · Zbl 0996.60001 · doi:10.1007/978-1-4757-4015-8 |
| [25] | Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York. · Zbl 0734.60060 · doi:10.1007/978-1-4612-0949-2 |
| [26] | Kenyon, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math. 185 239-286. · Zbl 0982.05013 · doi:10.1007/BF02392811 |
| [27] | Kesten, H. (1987). Hitting probabilities of random walks on \[{\mathbf{Z}^d} \]. Stochastic Process. Appl. 25 165-184. · Zbl 0626.60067 · doi:10.1016/0304-4149(87)90196-7 |
| [28] | Kigami, J. (1995). Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 48-86. · Zbl 0820.60060 · doi:10.1006/jfan.1995.1023 |
| [29] | Kigami, J. (2012). Resistance forms, quasisymmetric maps and heat kernel estimates. Mem. Amer. Math. Soc. 216 vi+132. · Zbl 1246.60099 · doi:10.1090/S0065-9266-2011-00632-5 |
| [30] | Kliem, S. and Löhr, W. (2015). Existence of mark functions in marked metric measure spaces. Electron. J. Probab. 20 73. · Zbl 1350.60103 · doi:10.1214/EJP.v20-3969 |
| [31] | Kozma, G. (2007). The scaling limit of loop-erased random walk in three dimensions. Acta Math. 199 29-152. · Zbl 1144.60060 · doi:10.1007/s11511-007-0018-8 |
| [32] | Kumagai, T. (2004). Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms. Publ. Res. Inst. Math. Sci. 40 793-818. · Zbl 1067.60070 |
| [33] | Kumagai, T. and Misumi, J. (2008). Heat kernel estimates for strongly recurrent random walk on random media. J. Theoret. Probab. 21 910-935. · Zbl 1159.60029 · doi:10.1007/s10959-008-0183-5 |
| [34] | Lawler, G. F. (1991). Intersections of Random Walks. Probability and Its Applications. Birkhäuser, Inc., Boston, MA. · Zbl 1228.60004 |
| [35] | Lawler, G. F. (1999). Loop-erased random walk. In Perplexing Problems in Probability. Progress in Probability 44 197-217. Birkhäuser, Boston, MA. · Zbl 0947.60055 |
| [36] | Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI. · Zbl 1074.60002 · doi:10.1090/surv/114 |
| [37] | Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge. · Zbl 1210.60002 · doi:10.1017/CBO9780511750854 |
| [38] | Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939-995. · Zbl 1126.82011 · doi:10.1214/aop/1079021469 |
| [39] | Le Gall, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 35-62. · Zbl 1129.60047 |
| [40] | Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. With a chapter by James G. Propp and David B. Wilson. · Zbl 1160.60001 · doi:10.1090/mbk/058 |
| [41] | Li, X. and Shiraishi, D. (2018). Convergence of three-dimensional loop-erased random walk in the natural parametrization. Preprint. Available at arXiv:1811.11685. |
| [42] | Li, X. and Shiraishi, D. (2019). One-point function estimates for loop-erased random walk in three dimensions. Electron. J. Probab. 24 111. · Zbl 1431.82025 · doi:10.1214/19-ejp361 |
| [43] | Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics 42. Cambridge Univ. Press, New York. · Zbl 1376.05002 · doi:10.1017/9781316672815 |
| [44] | Lyons, R., Peres, Y. and Schramm, O. (2003). Markov chain intersections and the loop-erased walk. Ann. Inst. Henri Poincaré Probab. Stat. 39 779-791. · Zbl 1030.60035 · doi:10.1016/S0246-0203(03)00033-5 |
| [45] | Masson, R. (2009). The growth exponent for planar loop-erased random walk. Electron. J. Probab. 14 1012-1073. · Zbl 1191.60061 · doi:10.1214/EJP.v14-651 |
| [46] | Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559-1574. · Zbl 0758.60010 |
| [47] | Sapozhnikov, A. and Shiraishi, D. (2018). On Brownian motion, simple paths, and loops. Probab. Theory Related Fields 172 615-662. · Zbl 1404.60062 · doi:10.1007/s00440-017-0817-6 |
| [48] | Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093 · doi:10.1007/BF02803524 |
| [49] | Shiraishi, D. (2018). Growth exponent for loop-erased random walk in three dimensions. Ann. Probab. 46 687-774. · Zbl 1387.60067 · doi:10.1214/16-AOP1165 |
| [50] | Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239-244. · Zbl 0985.60090 · doi:10.1016/S0764-4442(01)01991-7 |
| [51] | Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996) 296-303. ACM, New York. · Zbl 0946.60070 · doi:10.1145/237814.237880 |
| [52] | Wilson, D. B. (2010). Dimension of loop-erased random walk in three dimensions. Phys. Rev. E 82 062102 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
