Book Review
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Original version posted August 9, 2017
Corrected version posted October 13, 2017:
Current version corrects the use of an incorrect symbol in the original, in the displayed equations on page 3, first displayed equation, and page 4, displayed equations 3 and 4.
MathSciNet review:
3738540
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Book Information:
Authors:
Christophe Garban and
Jeffrey E. Steif
Title:
Noise sensitivity of Boolean functions and percolation
Additional book information:
Institute of Mathematical Statistics Textbooks, Vol. 5,
Cambridge University Press,
New York,
2015,
xvii+203 pp.,
ISBN 978-1-107-07643-3
Antonio Auffinger, Jack Hanson, and Michael Damron, 50 years of first passage percolation, 2015. arXiv:1511.03262, 50.
Michael Ben-Or and Nathan Linial, Collective coin flipping. Leibniz Center for Research in Computer Science, Department of Computer Science, Hebrew University of Jerusalem, 1987.
Itai Benjamini, Gil Kalai, and Oded Schramm, Noise sensitivity of Boolean functions and applications to percolation, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 5–43 (2001). MR 1813223
Itai Benjamini, Gil Kalai, and Oded Schramm, First passage percolation has sublinear distance variance [MR2016607], Selected works of Oded Schramm. Volume 1, 2, Sel. Works Probab. Stat., Springer, New York, 2011, pp. 779–787. MR 2883392, DOI 10.1007/978-1-4419-9675-6_{2}6
Marek Biskup, Oren Louidor, Eviatar B. Procaccia, and Ron Rosenthal, Isoperimetry in two-dimensional percolation, Comm. Pure Appl. Math. 68 (2015), no. 9, 1483–1531. MR 3378192, DOI 10.1002/cpa.21558
Avrim Blum, Ioannis Caragiannis, Nika Haghtalab, Ariel D Procaccia, Eviatar B Procaccia, and Rohit Vaish, Opting into optimal matchings, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2351–2363, SIAM, Philadelphia, PA, 2017.
Sourav Chatterjee, Superconcentration and related topics, Springer Monographs in Mathematics, Springer, Cham, 2014. MR 3157205, DOI 10.1007/978-3-319-03886-5
Michael Damron, Jack Hanson, and Philippe Sosoe, Sublinear variance in first-passage percolation for general distributions, Probab. Theory Related Fields 163 (2015), no. 1-2, 223–258. MR 3405617, DOI 10.1007/s00440-014-0591-7
Julian Gold, Isoperimetry in supercritical bond percolation in dimensions three and higher. arXiv:1602.05598, 2016.
Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688
Jeff Kahn, Gil Kalai, and Nathan Linial, The influence of variables on boolean functions, in 29th Annual Symposium on Foundations of Computer Science, 1988, pp. 68–80, IEEE, 1988.
Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), no. 9, 889.
Nathan Keller, Elchanan Mossel, and Arnab Sen, Geometric influences, Ann. Probab. 40 (2012), no. 3, 1135–1166. MR 2962089, DOI 10.1214/11-AOP643
Nathan Keller, Elchanan Mossel, and Arnab Sen, Geometric influences II: correlation inequalities and noise sensitivity, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1121–1139 (English, with English and French summaries). MR 3269987, DOI 10.1214/13-AIHP557
Harry Kesten, The critical probability of bond percolation on the square lattice equals ${1\over 2}$, Comm. Math. Phys. 74 (1980), no. 1, 41–59. MR 575895
J. F. C. Kingman, Subadditive ergodic theory, Ann. Probability 1 (1973), 883–909. MR 356192, DOI 10.1214/aop/1176996798
Thomas M. Liggett, An improved subadditive ergodic theorem, Ann. Probab. 13 (1985), no. 4, 1279–1285. MR 806224
Edward Nelson, The free Markoff field, J. Functional Analysis 12 (1973), 211–227. MR 0343816, DOI 10.1016/0022-1236(73)90025-6
Eviatar B. Procaccia and Ron Rosenthal, Concentration estimates for the isoperimetric constant of the supercritical percolation cluster, Electron. Commun. Probab. 17 (2012), no. 30, 11. MR 2955495, DOI 10.1214/ECP.v17-2185
Michel Talagrand, On Russo’s approximate zero-one law, Ann. Probab. 22 (1994), no. 3, 1576–1587. MR 1303654
References
- Antonio Auffinger, Jack Hanson, and Michael Damron, 50 years of first passage percolation, 2015. arXiv:1511.03262, 50.
- Michael Ben-Or and Nathan Linial, Collective coin flipping. Leibniz Center for Research in Computer Science, Department of Computer Science, Hebrew University of Jerusalem, 1987.
- Itai Benjamini, Gil Kalai, and Oded Schramm, Noise sensitivity of Boolean functions and applications to percolation, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 5–43 (2001). MR 1813223
- Itai Benjamini, Gil Kalai, and Oded Schramm, First passage percolation has sublinear distance variance [MR2016607], Selected Works of Oded Schramm. Volume 1, 2, Sel. Works Probab. Stat., Springer, New York, 2011, pp. 779–787. MR 2883392, DOI 10.1007/978-1-4419-9675-6_26
- Marek Biskup, Oren Louidor, Eviatar B. Procaccia, and Ron Rosenthal, Isoperimetry in two-dimensional percolation, Comm. Pure Appl. Math. 68 (2015), no. 9, 1483–1531. MR 3378192, DOI 10.1002/cpa.21558
- Avrim Blum, Ioannis Caragiannis, Nika Haghtalab, Ariel D Procaccia, Eviatar B Procaccia, and Rohit Vaish, Opting into optimal matchings, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2351–2363, SIAM, Philadelphia, PA, 2017.
- Sourav Chatterjee, Superconcentration and related topics, Springer Monographs in Mathematics, Springer, Cham, 2014. MR 3157205, DOI 10.1007/978-3-319-03886-5
- Michael Damron, Jack Hanson, and Philippe Sosoe, Sublinear variance in first-passage percolation for general distributions, Probab. Theory Related Fields 163 (2015), no. 1-2, 223–258. MR 3405617, DOI 10.1007/s00440-014-0591-7
- Julian Gold, Isoperimetry in supercritical bond percolation in dimensions three and higher. arXiv:1602.05598, 2016.
- Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 0420249, DOI 10.2307/2373688
- Jeff Kahn, Gil Kalai, and Nathan Linial, The influence of variables on boolean functions, in 29th Annual Symposium on Foundations of Computer Science, 1988, pp. 68–80, IEEE, 1988.
- Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), no. 9, 889.
- Nathan Keller, Elchanan Mossel, and Arnab Sen, Geometric influences, Ann. Probab. 40 (2012), no. 3, 1135–1166. MR 2962089, DOI 10.1214/11-AOP643
- Nathan Keller, Elchanan Mossel, and Arnab Sen, Geometric influences II: Correlation inequalities and noise sensitivity, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1121–1139 (English, with English and French summaries). MR 3269987, DOI 10.1214/13-AIHP557
- Harry Kesten, The critical probability of bond percolation on the square lattice equals ${1\over 2}$, Comm. Math. Phys. 74 (1980), no. 1, 41–59. MR 575895
- J. F. C. Kingman, Subadditive ergodic theory, Ann. Probab. 1 (1973), 883–909. With discussion by D. L. Burkholder, Daryl Daley, H. Kesten, P. Ney, Frank Spitzer and J. M. Hammersley, and a reply by the author. MR 0356192
- Thomas M. Liggett, An improved subadditive ergodic theorem, Ann. Probab. 13 (1985), no. 4, 1279–1285. MR 806224
- Edward Nelson, The free Markoff field, J. Funct. Anal. 12 (1973), 211–227. MR 0343816
- Eviatar B. Procaccia and Ron Rosenthal, Concentration estimates for the isoperimetric constant of the supercritical percolation cluster, Electron. Commun. Probab. 17 (2012), no. 30, 11. MR 2955495, DOI 10.1214/ECP.v17-2185
- Michel Talagrand, On Russo’s approximate zero-one law, Ann. Probab. 22 (1994), no. 3, 1576–1587. MR 1303654
Review Information:
Reviewer:
Eviatar B. Procaccia
Affiliation:
Department of Mathematics, Texas A&M University
Email:
eviatarp@gmail.com
Journal:
Bull. Amer. Math. Soc.
55 (2018), 131-138
DOI:
https://doi.org/10.1090/bull/1591
Published electronically:
August 9, 2017
Review copyright:
© Copyright 2017
American Mathematical Society
