Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of stochastic processes associated with resistance forms

D. A. Croydon

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We establish that if a sequence of spaces equipped with resistance metrics and measures converge with respect to the Gromov–Hausdorff-vague topology, and a certain non-explosion condition is satisfied, then the associated stochastic processes also converge. This result generalises previous work on trees, fractals, and various models of random graphs. We further conjecture that it will be applicable to the random walk on the incipient infinite cluster of critical bond percolation on the high-dimensional integer lattice.


Nous établissons que si une suite d’espaces équipée des métriques de résistance et de mesures converge par rapport à la topologie de Gromov–Hausdorff-vague, et qu’une certaine condition de non explosion est satisfaite, alors les processus stochastiques associés convergent également. Ces résultats généralisent des travaux précédents sur les arbres, fractals et divers modèles de graphes aléatoires. De plus nous conjecturons que cela devrait s’appliquer à la marche aléatoire sur l’amas de percolation par arêtes au point critique conditionné à être infini sur les réseaux entiers de grande dimension.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 1939-1968.

Received: 23 January 2017
Revised: 29 May 2017
Accepted: 21 August 2017
First available in Project Euclid: 18 October 2018

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Primary: 60J25: Continuous-time Markov processes on general state spaces 28A80: Fractals [See also 37Fxx] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Fractal Gromov–Hausdorff-vague topology Random graph Resistance form Resolvent kernel Tree


Croydon, D. A. Scaling limits of stochastic processes associated with resistance forms. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 1939--1968. doi:10.1214/17-AIHP861.

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  • [1] L. Addario-Berry, N. Broutin and C. Goldschmidt. The continuum limit of critical random graphs. Probab. Theory Related Fields 152 (3–4) (2012) 367–406.
  • [2] L. Addario-Berry, N. Broutin, C. Goldschmidt and G. Miermont. The scaling limit of the minimum spanning tree of the complete graph. Ann. Probab. To appear.
  • [3] S. Athreya, M. Eckhoff and A. Winter. Brownian motion on $\mathbb{R}$-trees. Trans. Amer. Math. Soc. 365 (6) (2013) 3115–3150.
  • [4] S. Athreya, W. Löhr and A. Winter. The gap between Gromov-vague and Gromov–Hausdorff-vague topology. Stochastic Process. Appl. 126 (9) (2016) 2527–2553.
  • [5] S. Athreya, W. Löhr and A. Winter. Invariance principle for variable speed random walks on trees. Ann. Probab. To appear.
  • [6] 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. To appear.
  • [7] G. Ben Arous, M. Cabezas and A. Fribergh. Scaling limit for the ant in a simple labyrinth. Preprint. Available at arXiv:1609.03980.
  • [8] G. Ben Arous, M. Cabezas and A. Fribergh. Scaling limit for the ant in high-dimensional labyrinths. Preprint. Available at arXiv:1609.03977.
  • [9] G. Ben Arous, M. Cabezas and A. Fribergh. Scaling limits for random walks on random critical trees. Preprint. Available at arXiv:1705.05883.
  • [10] I. Benjamini, O. Gurel-Gurevich and R. Lyons. Recurrence of random walk traces. Ann. Probab. 35 (2) (2007) 732–738.
  • [11] D. Burago, Y. Burago and S. Ivanov. A Course in Metric Geometry. Graduate Studies in Mathematics 33. Am. Math. Soc., Providence, 2001.
  • [12] D. A. Croydon. Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. Henri Poincaré Probab. Stat. 44 (6) (2008) 987–1019.
  • [13] D. A. Croydon. Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. Ann. Probab. 37 (3) (2009) 946–978.
  • [14] D. A. Croydon. Random walk on the range of random walk. J. Stat. Phys. 136 (2) (2009) 349–372.
  • [15] D. A. Croydon. Scaling limits for simple random walks on random ordered graph trees. Adv. in Appl. Probab. 42 (2) (2010) 528–558.
  • [16] D. A. Croydon. Scaling limit for the random walk on the largest connected component of the critical random graph. Publ. Res. Inst. Math. Sci. 48 (2) (2012) 279–338.
  • [17] D. A. Croydon, B. M. Hambly and T. Kumagai. Convergence of mixing times for sequences of random walks on finite graphs. Electron. J. Probab. 17 (3) (2012) 32.
  • [18] D. A. Croydon, B. M. Hambly and T. Kumagai. Time-changes of stochastic processes associated with resistance forms. Preprint. Available at arXiv:1609.02120.
  • [19] T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (4) (2005) 553–603.
  • [20] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes, extended edition. de Gruyter Studies in Mathematics 19. de Gruyter, Berlin, 2011.
  • [21] G. Georganopoulos. Sur l’approximation des fonctions continues par des fonctions lipschitziennes. C. R. Acad. Sci. Paris Sér. A-B 264 (1967) A319–A321.
  • [22] A. L. Gibbs and F. E. Su. On choosing and bounding probability metrics. Int. Stat. Rev. 70 (3) (2002) 419–435.
  • [23] N. Gigli, A. Mondino and G. Savaré. Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows. Proc. Lond. Math. Soc. (3) 111 (5) (2015) 1071–1129.
  • [24] B. Haas and G. Miermont. Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 (6) (2012) 2589–2666.
  • [25] B. M. Hambly and W. Yang. Degenerate limits for one-parameter families of non-fixed-point diffusions on fractals. Preprint. Available at arXiv:1612.02342.
  • [26] T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys. 41 (3) (2000) 1244–1293.
  • [27] M. Heydenreich, R. van der Hofstad and T. Hulshof. Random walk on the high-dimensional IIC. Comm. Math. Phys. 329 (1) (2014) 57–115.
  • [28] N. Holden and X. Sun. SLE as a mating of trees in Euclidean geometry. Preprint. Available at arXiv:1610.05272.
  • [29] S. Janson and J.-F. Marckert. Convergence of discrete snakes. J. Theoret. Probab. 18 (3) (2005) 615–647.
  • [30] N. Kajino. Neumann and Dirichlet heat kernel estimates in inner uniform domains for local resistance forms. In preparation.
  • [31] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer, New York, 2002.
  • [32] H. Kesten. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22 (4) (1986) 425–487.
  • [33] J. Kigami. Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 (1) (1995) 48–86.
  • [34] J. Kigami. Analysis on Fractals. Cambridge Tracts in Mathematics 143. Cambridge University Press, Cambridge, 2001.
  • [35] J. Kigami. Resistance forms, quasisymmetric maps and heat kernel estimates. Mem. Amer. Math. Soc. 216, no. 1015, vi+132 (2012).
  • [36] T. Kumagai. Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms. Publ. Res. Inst. Math. Sci. 40 (3) (2004) 793–818.
  • [37] M. B. Marcus and J. Rosen. Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge University Press, Cambridge, 2006.
  • [38] G. Miermont. Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 (5) (2009) 725–781.
  • [39] C. Stone. Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 (1963) 638–660.
  • [40] K. Suzuki. Convergence of Brownian motions on RCD∗(K, $\infty$) spaces. Preprint. Available at arXiv:1603.08622.
  • [41] K. Suzuki. Convergence of Brownian motions on RCD∗(K, N) spaces. Preprint. Available at arXiv:1509.02025.
  • [42] M. Talagrand. Regularity of Gaussian processes. Acta Math. 159 (1–2) (1987) 99–149.
  • [43] R. van der Hofstad and A. A. Járai. The incipient infinite cluster for high-dimensional unoriented percolation. J. Stat. Phys. 114 (3–4) (2004) 625–663.