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, DOI 10.1017/9781107415690
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, DOI 10.1017/CBO9780511750854
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, DOI 10.1090/mbk/107
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, DOI 10.1017/9781316672815
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., Graduate Texts in Mathematics, Vol. 34, Springer-Verlag, New York-Heidelberg, 1976. 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 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 10.1016/0022-1236(85)90086-2
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
References
- 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, DOI 10.1017/9781107415690
- 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, DOI 10.1017/CBO9780511750854
- David A. Levin and Yuval Peres, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2017. 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, DOI 10.1017/9781316672815
- 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 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 10.1016/0022-1236(85)90086-2
- 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
