## The Annals of Probability

- Ann. Probab.
- Volume 39, Number 5 (2011), 1702-1767.

### Oded Schramm’s contributions to noise sensitivity

**Full-text: Open access**

#### Abstract

We survey in this paper the main contributions of Oded Schramm related to *noise sensitivity*. We will describe in particular his various works which focused on the “spectral analysis” of critical percolation (and more generally of Boolean functions), his work on the shape-fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation.

#### Article information

**Source**

Ann. Probab. Volume 39, Number 5 (2011), 1702-1767.

**Dates**

First available in Project Euclid: 18 October 2011

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1318940780

**Digital Object Identifier**

doi:10.1214/10-AOP582

**Mathematical Reviews number (MathSciNet)**

MR2884872

**Zentralblatt MATH identifier**

1252.82090

**Subjects**

Primary: 82C43: Time-dependent percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 42B05: Fourier series and coefficients

**Keywords**

Percolation noise sensitivity discrete Fourier analysis hypercontractivity randomized algorithms SLE processes critical exponents first-passage percolation sub-Gaussian fluctuations

#### Citation

Garban, Christophe. Oded Schramm’s contributions to noise sensitivity. Ann. Probab. 39 (2011), no. 5, 1702--1767. doi:10.1214/10-AOP582. http://projecteuclid.org/euclid.aop/1318940780.

#### References

- [1] Ben-Or, M. and Linial, N. (1987). Collective coin flipping. In
*Randomness and Computation*(S. Micali, ed.). Academic Press, New York. - [2] Benaïm, M. and Rossignol, R. (2008). Exponential concentration for first passage percolation through modified Poincaré inequalities.
*Ann. Inst. H. Poincaré Probab. Statist.***44**544–573. - [3] Benjamini, I., Kalai, G. and Schramm, O. (1999). Noise sensitivity of Boolean functions and applications to percolation.
*Publ. Math. Inst. Hautes Études Sci.***90**5–43. - [4] Benjamini, I., Kalai, G. and Schramm, O. (2003). First passage percolation has sublinear distance variance.
*Ann. Probab.***31**1970–1978. - [5] Benjamini, I. and Schramm, O. (1998). Conformal invariance of Voronoi percolation.
*Comm. Math. Phys.***197**75–107. - [6] Benjamini, I. and Schramm, O. (1998). Exceptional planes of percolation.
*Probab. Theory Related Fields***111**551–564. - [7] Benjamini, I., Schramm, O. and Wilson, D. B. (2005). Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read. In
*STOC’*05:*Proceedings of the*37*th Annual ACM Symposium on Theory of Computing*244–250. ACM, New York. - [8] Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y. and Linial, N. (1992). The influence of variables in product spaces.
*Israel J. Math.***77**55–64. - [9] Cardy, J. L. (1992). Critical percolation in finite geometries.
*J. Phys. A***25**L201–L206. - [10] Chatterjee, S. (2008). Chaos, concentration, and multiple valleys. Preprint.
- [11] Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold.
*Proc. Amer. Math. Soc.***124**2993–3002. - [12] Garban, C., Pete, G. and Schramm, O. (2010). The Fourier spectrum of critical percolation.
*Acta Math.***205**19–104. - [13] Grimmett, G. (1999).
*Percolation*, 2nd ed.*Grundlehren der Mathematischen Wissenschaften*[*Fundamental Principles of Mathematical Sciences*]**321**. Springer, Berlin. - [14] Grimmett, G. (2008).
*Probability on Graphs*. Cambridge Univ. Press, Cambridge. - [15] Gross, L. (1975). Logarithmic Sobolev inequalities.
*Amer. J. Math.***97**1061–1083. - [16] Häggström, O., Peres, Y. and Steif, J. E. (1997). Dynamical percolation.
*Ann. Inst. H. Poincaré Probab. Statist.***33**497–528. - [17] Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions.
*Comm. Math. Phys.***128**333–391. - [18] Johansson, K. (2000). Shape fluctuations and random matrices.
*Comm. Math. Phys.***209**437–476. - [19] Kahn, J., Kalai, G. and Linial, N. (1988). The influence of variables on Boolean functions. In 29
*th Annual Symposium on Foundations of Computer Science*68–80. IEEE Computer Society, Washington, DC. - [20] Kalai, G. and Safra, S. (2006). Threshold phenomena and influence: Perspectives from mathematics, computer science, and economics. In
*Computational Complexity and Statistical Physics*25–60. Oxford Univ. Press, New York. - [21] Kesten, H. (1987). Scaling relations for 2D-percolation.
*Comm. Math. Phys.***109**109–156. - [22] Kesten, H. and Zhang, Y. (1987). Strict inequalities for some critical exponents in two-dimensional percolation.
*J. Stat. Phys.***46**1031–1055. - [23] Langlands, R., Pouliot, P. and Saint-Aubin, Y. (1994). Conformal invariance in two-dimensional percolation.
*Bull. Amer. Math. Soc.*(*N.S.*)**30**1–61. - [24] Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation.
*Electron. J. Probab.***7**13 pp. (electronic). - [25] Margulis, G. A. (1974). Probabilistic characteristics of graphs with large connectivity.
*Problemy Peredači Informacii***10**101–108. - [26] Meester, R. and Roy, R. (1996).
*Continuum Percolation. Cambridge Tracts in Mathematics***119**. Cambridge Univ. Press, Cambridge. - [27] Mossel, E., O’Donnell, R. and Oleszkiewicz, K. (2010). Noise stability of functions with low influences: Invariance and optimality.
*Ann. of Math.*(2)**171**295–341. - [28] Nelson, E. (1966). A quartic interaction in two dimensions. In
*Mathematical Theory of Elementary Particles*(*Proc. Conf.*,*Dedham*,*Mass.*, 1965) 69–73. MIT Press, Cambridge, MA. - [29] Nolin, P. (2008). Near-critical percolation in two dimensions.
*Electron. J. Probab.***13**1562–1623. - [30] O’Donnell, R. History of the hypercontractivity theorem. Available at http://boolean-analysis.blogspot.com/.
- [31] O’Donnell, R. W. (2003). Computational applications of noise sensitivity. Ph.D. thesis, MIT.
- [32] O’Donnell, R., Saks, M., Schramm, O. and Servedio, R. A. (2005). Every decision tree has an influential variable. In 46
*th Annual IEEE Symposium on Foundations of Computer Science*(*FOCS’*05) 31–39. IEEE Computer Society, Los Alamitos, CA. - [33] O’Donnell, R. and Servedio, R. A. (2007). Learning monotone decision trees in polynomial time.
*SIAM J. Comput.***37**827–844 (electronic). - [34] Peres, Y., Schramm, O., Sheffield, S. and Wilson, D. B. (2007). Random-turn hex and other selection games.
*Amer. Math. Monthly***114**373–387. - [35] Russo, L. (1978). A note on percolation.
*Z. Wahrsch. Verw. Gebiete***43**39–48. - [36] Russo, L. (1982). An approximate zero-one law.
*Z. Wahrsch. Verw. Gebiete***61**129–139. - [37] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees.
*Israel J. Math.***118**221–288. - [38] Schramm, O. (2007). Conformally invariant scaling limits: An overview and a collection of problems. In
*International Congress of Mathematicians***I**513–543. Eur. Math. Soc., Zürich. - [39] Schramm, O. and Smirnov, S. (2011). On the scaling limits of planar percolation.
*Ann. Probab.***39**1768–1814. - [40] Schramm, O. and Steif, J. E. (2010). Quantitative noise sensitivity and exceptional times for percolation.
*Ann. of Math.*(2)**171**619–672. - [41] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits.
*C. R. Acad. Sci. Paris Sér. I Math.***333**239–244. - [42] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation.
*Math. Res. Lett.***8**729–744. - [43] Steif, J. (2009). A survey of dynamical percolation. In
*Fractal Geometry and Stochastics***IV**145–174. Birkhäuser, Berlin. - [44] Talagrand, M. (1994). On Russo’s approximate zero-one law.
*Ann. Probab.***22**1576–1587. - [45] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces.
*Inst. Hautes Études Sci. Publ. Math.***81**73–205. - [46] Talagrand, M. (1996). How much are increasing sets positively correlated?
*Combinatorica***16**243–258. - [47] Tsirelson, B. (1999). Fourier–Walsh coefficients for a coalescing flow (discrete time).
- [48] Tsirelson, B. (2004). Scaling limit, noise, stability. In
*Lectures on Probability Theory and Statistics. Lecture Notes in Math.***1840**1–106. Springer, Berlin. - [49] Tsirelson, B. S. and Vershik, A. M. (1998). Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations.
*Rev. Math. Phys.***10**81–145. - [50] Werner, W. (2007). Lectures on two-dimensional critical percolation. IAS Park City Graduate Summer School.

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