Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Random Walks and Heat Kernels on Graphs
  • Cited by 59
Publisher:
Cambridge University Press
Online publication date:
March 2017
Print publication year:
2017
Online ISBN:
9781107415690
DOI:
https://doi.org/10.1017/9781107415690
Subjects:
Mathematics (general), Mathematics, Abstract Analysis, Probability Theory and Stochastic Processes, Statistics and Probability
Series:
London Mathematical Society Lecture Note Series (438)
Information

Book description

This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Poincaré inequalities. The book presents rough isometries and looks at the properties of a graph that are stable under these transformations. Applications include the 'type problem': determining whether a graph is transient or recurrent. The final chapters show how geometric properties of the graph can be used to establish heat kernel bounds, that is, bounds on the transition probabilities of the random walk, and it is proved that Gaussian bounds hold for graphs that are roughly isometric to Euclidean space. Aimed at graduate students in mathematics, the book is also useful for researchers as a reference for results that are hard to find elsewhere.

Reviews

'This book, written with great care, is a comprehensive course on random walks on graphs, with a focus on the relation between rough geometric properties of the underlying graph and the asymptotic behavior of the random walk on it. It is accessible to graduate students but may also serve as a good reference for researchers. It contains the usual material about random walks on graphs and its connections to discrete potential theory and electrical resistance (Chapters 1, 2 and 3). The heart of the book is then devoted to the study of the heat kernel (Chapters 4, 5 and 6). The author develops sufficient conditions under which sub-Gaussian or Gaussian bounds for the heat kernel hold (both on-diagonal and off diagonal; both upper and lower bounds).'

Nicolas Curien Source: Mathematical Review

'The book under review delineates very thoroughly the general theory of random walks on weighted graphs. The author’s expertise in both probability and analysis is apparent in the exposition and the elegant proofs depicted in the book.'

Eviatar B. Procaccia Source: Bulletin of the American Mathematical Society

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
References
[AF] D.J., Aldous and J., Fill Google Scholar. Reversible Markov Chains and Random Walks on Graphs. See http://www.stat.berkeley.edu/∼aldous/RWG/book.html
[Ah] L.V., Ahlfors. Conformal Invariants: Topics in Geometric Function Theory. New York, NY: McGraw-Hill Book Co. (1973 Google Scholar).
[Ar] D.G., Aronson. Bounds on the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 (1967 Google Scholar) 890–896.
[Ba1] M.T., Barlow Which values of the volume growth and escape time exponent are possible for a graph? Rev. Mat. Iberoamer. 20 (2004 Google Scholar), 1–31.
[Ba2] M.T., Barlow. Some remarks on the elliptic Harnack inequality. Bull. London Math. Soc. 37 (2005 Google Scholar), no. 2, 200–208.
[BB1] M.T., Barlow and R.F., Bass. Random walks on graphical Sierpinski carpets. In: Random Walks and Discrete Potential Theory, eds. M., Piccardello and W., Woess. Symposia Mathematica XXXIX. Cambridge: Cambridge University Press (1999 Google Scholar).
[BB2] M.T., Barlow and R.F., Bass. Stability of parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356 (2003 Google Scholar), no. 4, 1501–1533.
[BC] M.T., Barlow and X., Chen. Gaussian bounds and parabolic Harnack inequality on locally irregular graphs. Math. Annalen 366 (2016 Google Scholar), no. 3, 1677–1720.
[BCK] M.T., Barlow, T., Coulhon, and T., Kumagai. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Comm. Pure. Appl. Math. LVIII (2005 Google Scholar), 1642–1677.
[BH] M.T., Barlow and B.M., Hambly. Parabolic Harnack inequality and local limit theorem for percolation clusters. Electron. J. Prob. 14 (2009 Google Scholar), no. 1, 1–27.
[BK] M.T., Barlow and T., Kumagai. Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 (Doob volume) (2006 Google Scholar), 33–65.
[BJKS] M.T., Barlow, A.A., Járai, T., Kumagai and G., Slade. Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Commun. Math. Phys. 278 (2008 Google Scholar), no. 2, 385–431.
[BM] M.T., Barlow and R., Masson. Spectral dimension and random walks on the two dimensional uniform spanning tree. Commun. Math. Phys. 305 (2011 Google Scholar), no. 1, 23–57.
[BMu] M.T., Barlow and M., Murugan Google Scholar. Stability of elliptic Harnack inequality. In prep.
[BT] Zs., Bartha and A., Telcs. Quenched invariance principle for the random walk on the Penrose tiling. Markov Proc. Related Fields 20 (2014 Google Scholar), no. 4, 751–767.
[Bas] R.F., Bass. On Aronsen's upper bounds for heat kernels. Bull. London Math. Soc. 34 (2002 Google Scholar), 415–419.
[BPP] I., Benjamini, R., Pemantle, and Y., Peres. Unpredictable paths and percolation. Ann. Prob. 26 (1998 Google Scholar), 1198–1211.
[BD] A., Beurling and J., Deny. Espaces de Dirichlet. I. Le cas élémentaire. Acta Math. 99 (1958 Google Scholar), 203–224.
[BL] K., Burdzy and G.F., Lawler. Rigorous exponent inequalities for random walks. J. Phys. A 23 (1990 Google Scholar), L23–L28.
[Bov] A., Bovier. Metastability: a potential theoretic approach. Proc. ICM Madrid 2006. Zurich Google Scholar: European Mathematical Society.
[CKS] E.A., Carlen, S., Kusuoka, and D.W., Stroock. Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Suppl. no. 2 (1987 Google Scholar), 245– 287.
[Ca] T.K., Carne. A transmutation formula for Markov chains. Bull. Sci. Math. 109 (1985 Google Scholar), 399–405.
[CRR] A.K., Chandra, P., Raghavan, W.L., Ruzzo, R., Smolensky, and P., Tiwari. The electrical resistance of a graph captures its commute and cover times. In: Proc. 21st Annual ACM Symposium on Theory of Computing, Seattle, WA, ed. D.S., Johnson. New York, NY: ACM (1989 Google Scholar).
[Ch] X., Chen Google Scholar. Pointwise upper estimates for transition probability of continuous time random walks on graphs. In prep.
[Co1] T., Coulhon. Espaces de Lipschitz et inégalités de Poincaré. J. Funct. Anal. 136 (1996 Google Scholar), no. 1, 81–113.
[Co2] T., Coulhon. Heat kernel and isoperimetry on non-compact Riemannian manifolds. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), eds. P. Auscher, T. Coulhon, and A. Grigor'yan. Contemporary Mathematics vol. 338. Providence, RI: AMS (2003 Google Scholar), pp. 65–99.
[CG] T., Coulhon and A., Grigor'yan. Random walks on graphs with regular volume growth. GAFA 8 (1998 Google Scholar), 656–701.
[CGZ] T., Coulhon, A., Grigoryan, and F., Zucca. The discrete integral maximum principle and its applications. Tohoku Math. J. (2) 57 (2005 Google Scholar), no. 4, 559–587.
[CS] T., Coulhon and A., Sikora. Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem. Proc. Lond. Math. Soc. (3) 96 (2008 Google Scholar), no. 2, 507–544.
[Da] E.B., Davies. Large deviations for heat kernels on graphs. J. London Math. Soc.(2) 47 (1993 Google Scholar), 65–72.
[D1] T., Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Math. Iberoamer. 15 (1999 Google Scholar), 181–232.
[D2] T., Delmotte. Graphs between the elliptic and parabolic Harnack inequalities. Potential Anal. 16 (2002 Google Scholar), no. 2, 151–168.
[De] Y., Derriennic. Lois ‘zero ou deux’ pour les processus de Markov. Ann. Inst. Henri Poincaré Sec. B 12 (1976 Google Scholar), 111–129.
[DiS] P., Diaconis and D., Stroock. Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1 (1991 Google Scholar), 36–61.
[DS] P., Doyle and J.L., Snell. Random Walks and Electrical Networks. Washington D.C.: Mathematical Association of America (1984 Google Scholar). Arxiv:.PR/0001057.
[Duf] R.J., Duffin. The extremal length of a network. J. Math. Anal. Appl. 5 (1962 Google Scholar), 200–215.
[Dur] R., Durrett. Probability: Theory and Examples, 4th edn. Cambridge: Cambridge University Press (2010 Google Scholar).
[DK] B., Dyda and M., Kassmann. On weighted Poincaré inequalities. Ann. Acad. Sci. Fenn. Math. 38 (2013 Google Scholar), 721–726.
[DM] E.B., Dynkin and M.B., Maljutov. Random walk on groups with a finite number of generators. (In Russian.) Dokl. Akad. Nauk SSSR 137 (1961 Google Scholar), 1042–1045.
[FS] E.B., Fabes and D.W., Stroock. A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash. Arch. Mech. Rat. Anal. 96 (1986 Google Scholar), 327– 338.
[Fa] K., Falconer. Fractal Geometry. Chichester: Wiley (1990 Google Scholar).
[Fe] W., Feller. An Introduction to Probability Theory and its Applications. Vol. I, 3rd edn. New York, NY: Wiley (1968 Google Scholar).
[Fo1] M., Folz. Gaussian upper bounds for heat kernels of continuous time simple random walks. Elec. J. Prob. 16 (2011 Google Scholar), 1693–1722, paper 62.
[Fo2] M., Folz. Volume growth and stochastic completeness of graphs. Trans. Amer. Math. Soc. 366 (2014 Google Scholar), 2089–2119.
[Fos] F.G., Foster. On the stochastic matrices associated with certain queuing processes. Ann. Math. Statist. 24 (1953 Google Scholar), 355–360.
[FOT] M., Fukushima, Y., Oshima, and M., Takeda, Dirichlet Forms and Symmetric Markov Processes. Berlin: de Gruyter (1994 Google Scholar).
[Ga] R.J., Gardner. The Brunn-Minkowski inequality. Bull. AMS 39 (2002 Google Scholar), 355– 405.
[Gg1] A.A., Grigor'yan. The heat equation on noncompact Riemannian manifolds. Math. USSR Sbornik 72 (1992 Google Scholar), 47–77.
[Gg2] A.A., Grigor'yan. Heat kernel upper bounds on a complete non-compact manifold. Revista Math. Iberoamer. 10 (1994 Google Scholar), 395–452.
[GT1] A., Grigor'yan and A., Telcs. Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109 (2001 Google Scholar), 452–510.
[GT2] A., Grigor'yan and A., Telcs. Harnack inequalities and sub-Gaussian estimates for random walks. Math. Annal. 324 (2002 Google Scholar), no. 3, 521–556.
[GHM] A., Grigor'yan, X.-P., Huang, and J., Masamune. On stochastic completeness of jump processes. Math. Z. 271 (2012 Google Scholar), no. 3, 1211–1239.
[Grom1] M., Gromov. Groups of polynomial growth and expanding maps. Publ. Math. IHES 53 (1981 Google Scholar), 53–73.
[Grom2] M., Gromov. Hyperbolic groups. In: Essays in Group Theory, ed. S.M., Gersten. New York, NY: Springer (1987 Google Scholar), pp. 75–263.
[HSC] W., Hebisch and L., Saloff-Coste. Gaussian estimates for Markov chains and random walks on groups. Ann. Prob. 21 (1993 Google Scholar), 673–709.
[Je] D., Jerison. The weighted Poincaré inequality for vector fields satisfying Hörmander's condition. Duke Math. J. 53 (1986 Google Scholar), 503–523.
[Kai] V.A., Kaimanovich. Measure-theoretic boundaries of 0-2 laws and entropy. In: Harmonic Analysis and Discrete Potential Theory (Frascati, 1991). New York, NY: Plenum (1992 Google Scholar), pp. 145–180.
[Kan1] M., Kanai. Rough isometries and combinatorial approximations of geometries of non-compact riemannian manifolds. J. Math. Soc. Japan 37 (1985 Google Scholar), 391–413.
[Kan2] M., Kanai. Analytic inequalities, and rough isometries between non-compact reimannian manifolds. In: Curvature and Topology of Riemannian Manifolds. Proc. 17th Intl. Taniguchi Symp., Katata, Japan, eds. K., Shiohama, T., Sakai, and T., Sunada. Lecture Notes in Mathematics 1201. New York, NY: Springer (1986 Google Scholar), pp. 122–137.
[KSK] J.G., Kemeny, J.L., Snell, and A.W., Knapp. Denumerable Markov Chains. New York, NY: Springer (1976 Google Scholar).
[Ki] J., Kigami. Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128 (1995 Google Scholar), no. 1, 48–86.
[Kir1] G., Kirchhoff. Über die Auflösung der Gleichungen, auf welche man bei der Untersuchungen der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72 (1847 Google Scholar), 497–508.
[Kir2] G., Kirchhoff. On the solution of the equations obtained from the investigation of the linear distribution of galvanic currents. (Translated by J.B. O'Toole.) IRE Trans. Circuit Theory 5 (1958 Google Scholar), 4–7.
[KN] G., Kozma and A., Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178 (2009 Google Scholar), no. 3, 635–654.
[Kum] T., Kumagai. Random Walks on Disordered Media and their Scaling Limits. École d'Été de Probabilités de Saint-Flour XL – 2010. Lecture Notes in Mathematics 2101. Chan: Springer International (2014 Google Scholar).
[LL] G.F., Lawler and V., Limic. Random Walk: A Modern Introduction. Cambridge: Cambridge University Press (2010 Google Scholar).
[LP] R., Lyons and Y., Peres. Probability on Trees and Networks. Cambridge: Cambridge University Press, 2016 Google Scholar.
[Ly1] T., Lyons. A simple criterion for transience of a reversible Markov chain. Ann. Prob. 11 (1983 Google Scholar), 393–402.
[Ly2] T., Lyons. Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Diff. Geom. 26 (1987 Google Scholar), 33–66.
[Mer] J.-F., Mertens, E., Samuel-Cahn, and S., Zamir. Necessary and sufficient conditions for recurrence and transience of Markov chains in terms of inequalities. J. Appl. Prob. 15 (1978 Google Scholar), 848–851.
[Mo1] J., Moser. On Harnack's inequality for elliptic differential equations. Commun. Pure Appl. Math. Google Scholar 14 (1961), 577–591.
[Mo2] J., Moser. On Harnack's inequality for parabolic differential equations. Commun. Pure Appl. Math. 17 (1964 Google Scholar), 101–134.
[Mo3] J., Moser. On a pointwise estimate for parabolic differential equations. Commun. Pure Appl. Math. 24 (1971 Google Scholar), 727–740.
[N] J., Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958 Google Scholar), 931–954.
[Nor] J., Norris. Markov Chains. Cambridge: Cambridge University Press (1998 Google Scholar).
[NW] C., St J.A., Nash-Williams. Random walks and electric currents in networks. Proc. Camb. Phil. Soc. 55 (1959 Google Scholar), 181–194.
[Os] H., Osada. Isoperimetric dimension and estimates of heat kernels of pre- Sierpinski carpets. Prob. Theor Related Fields 86 (1990 Google Scholar), 469–490.
[Pol] G., Polya. Über eine Ausgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann. 84 (1921 Google Scholar), 149–160.
[Rev] D., Revelle. Heat kernel asymptotics on the lamplighter group. Elec. Commun. Prob. 8 (2003 Google Scholar), 142–154.
[RW] L.C.G., Rogers and D.W., Williams. Diffusions, Markov Processes, and Martingales. Vol. 1. Foundations. 2nd edn. Chichester: Wiley (1994 Google Scholar).
[Ro] O.S., Rothaus. Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal. 64 (1985 Google Scholar), 296–313.
[SC1] L., Saloff-Coste. A note on Poincaré, Sobolev, and Harnack inequalities. Inter. Math. Res. Not 2 (1992 Google Scholar), 27–38.
[SC2] L., Saloff-Coste. Aspects of Sobolev-type Inequalities. LMS Lecture Notes 289. Cambridge: Cambridge University Press (2002 Google Scholar).
[SJ] A., Sinclair and M., Jerrum. Approximate counting, uniform generation and rapidly mixing Markov chains. Inform. Comput. 82 (1989 Google Scholar), 93–133.
[So1] P.M., Soardi. Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. AMS 119 (1993 Google Scholar), 1239–1248.
[So2] P.M., Soardi. Potential Theory on Infinite Networks. Berlin: Springer (1994 Google Scholar).
[Spi] F., Spitzer. Principles of Random Walk. New York, NY: Springer (1976 Google Scholar).
[SZ] D.W., Stroock and W., Zheng. Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Prob. Stat. 33 (1997 Google Scholar), 619–649.
[T1] A., Telcs. Random walks on graphs, electric networks and fractals. Prob. Theor. Related Fields 82 (1989 Google Scholar), 435–451.
[T2] A., Telcs. Local sub-Gaussian estimates on graphs: the strongly recurrent case. Electron. J. Prob. 6 (2001 Google Scholar), no. 22, 1–33.
[T3] A., Telcs. Diffusive limits on the Penrose tiling. J. Stat. Phys. 141 (2010 Google Scholar), 661–668.
[Tet] P., Tetali. Random walks and the effective resistance of networks. J. Theor. Prob. 4 (1991 Google Scholar), 101–109.
[Tru] K., Truemper. On the delta-wye reduction for planar graphs. J. Graph Theory 13 (1989 Google Scholar), 141–148.
[V1] N.Th., Varopoulos. Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63 (1985 Google Scholar), 215–239.
[V2] N.Th., Varopoulos. Long range estimates forMarkov chains. Bull. Sci. Math., 2e serie 109 (1985 Google Scholar), 225–252225–252.
[W] D., Williams. Probability with Martingales. Cambridge: Cambridge University Press (1991 Google Scholar).
[Wo] W., Woess. Random Walks on Infinite Graphs and Groups. Cambridge: Cambridge University Press (2000 Google Scholar).
[Z] A.H., Zemanian. Infinite Electrical Metworks. Cambridge: Cambridge University Press (1991 Google Scholar).
[Zie] W.P., Ziemer. Weakly Differentiable Functions. New York, NY: Springer (1989 Google Scholar).

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 5901 *
Loading metrics...

Book summary page views

Total views: 7543 *
Loading metrics...

* Views captured on Cambridge Core between 9th March 2017 - 4th April 2025. This data will be updated every 24 hours.

Usage data cannot currently be displayed.