Book Review

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MathSciNet review: 4007385

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Book Information:

Author: Martin T. Barlow

Title: Random walks and heat kernels on graphs

Additional book information: London Mathematical Society Lecture Notes Series, Vol. 438, Cambridge University Press, Cambridge, 2017, xi+226 pp., ISBN 978-1-107-67442-4, US$80

- Martin T. Barlow,
*Random walks and heat kernels on graphs*, London Mathematical Society Lecture Note Series, vol. 438, Cambridge University Press, Cambridge, 2017. MR**3616731** - Peter Buser,
*A note on the isoperimetric constant*, Ann. Sci. École Norm. Sup. (4)**15**(1982), no. 2, 213–230. MR**683635** - Thomas Keith Carne,
*A transmutation formula for Markov chains*, Bull. Sci. Math. (2)**109**(1985), no. 4, 399–405 (English, with French summary). MR**837740** - Jeff Cheeger,
*A lower bound for the smallest eigenvalue of the Laplacian*, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR**0402831** - Fan R. K. Chung,
*Spectral graph theory*, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR**1421568** - Peter G. Doyle and J. Laurie Snell,
*Random walks and electric networks*, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. MR**920811** - B. Hanin.
*Which neural net architectures give rise to exploding and vanishing gradients?*, arXiv:1801.03744 (2018). - Gregory F. Lawler and Vlada Limic,
*Random walk: a modern introduction*, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010. MR**2677157** - David A. Levin and Yuval Peres,
*Markov chains and mixing times*, American Mathematical Society, Providence, RI, 2017. Second edition of [ MR2466937]; With contributions by Elizabeth L. Wilmer; With a chapter on “Coupling from the past” by James G. Propp and David B. Wilson. MR**3726904** - Russell Lyons and Yuval Peres,
*Probability on trees and networks*, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016. MR**3616205** - K. Pearson,
*The problem of the random walk*, Nature (London)**72**(1905), 294. - J. W. Strutt, 3rd Baron Rayleigh,
*On the electromagnetic theory of light*, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science,**12**(1881), no 73, 81–101. - J. W. Strutt, 3rd Baron Rayleigh,
*The problem of random walk*, Nature (London)**72**(1905), 318–325. - Frank Spitzer,
*Principles of random walk*, 2nd ed., Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, Vol. 34. MR**0388547** - Toshikazu Sunada,
*Discrete geometric analysis*, Analysis on graphs and its applications, Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 51–83. MR**2459864**, DOI https://doi.org/10.1090/pspum/077/2459864 - N. Th. Varopoulos,
*Isoperimetric inequalities and Markov chains*, J. Funct. Anal.**63**(1985), no. 2, 215–239. MR**803093**, DOI https://doi.org/10.1016/0022-1236%2885%2990086-2 - Nicholas Th. Varopoulos,
*Long range estimates for Markov chains*, Bull. Sci. Math. (2)**109**(1985), no. 3, 225–252 (English, with French summary). MR**822826**

Review Information:

Reviewer: Eviatar B. Procaccia

Affiliation: Department of Mathematics, Texas A&M University

Email: procaccia@tamu.edu

Journal: Bull. Amer. Math. Soc.

**56**(2019), 705-711

DOI: https://doi.org/10.1090/bull/1643

Published electronically: August 3, 2018

Review copyright: © Copyright 2018 American Mathematical Society