## The Annals of Probability

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

### Oded Schramm’s contributions to noise sensitivity

**Full-text: Access denied (no subscription detected)** We're sorry, but we are unable to provide you with the
full text of this article because we are not able to identify you as
a subscriber. If you have a personal subscription to this journal,
then please login. If you are already logged in, then you may need
to update your profile to register your subscription. Read more about accessing full-text

#### 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: 18 October 2011

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/10-AOP582

**Zentralblatt MATH identifier**

05987668

**Mathematical Reviews number (MathSciNet)**

MR2884872

**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. The Annals of Probability 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.

### More like this

- Percolation beyond ℤd: The contributions of Oded Schramm

Häggström, Olle, The Annals of Probability, 2011 - Critical exponents of planar gradient percolation

Nolin, Pierre, The Annals of Probability, 2008 - Critical intensities of Boolean models with different underlying convex shapes

Roy, Rahul and Tanemura, Hideki, Advances in Applied Probability, 2002

- Percolation beyond ℤd: The contributions of Oded Schramm

Häggström, Olle, The Annals of Probability, 2011 - Critical exponents of planar gradient percolation

Nolin, Pierre, The Annals of Probability, 2008 - Critical intensities of Boolean models with different underlying convex shapes

Roy, Rahul and Tanemura, Hideki, Advances in Applied Probability, 2002 - Overshoots and undershoots of Lévy processes

Doney, R. A. and Kyprianou, A. E., The Annals of Applied Probability, 2006 - Percolation on a product of two trees

Kozma, Gady, The Annals of Probability, 2011 - Strict inequalities for the time constant in first passage percolation

Marchand, R., The Annals of Applied Probability, 2002 - J. K. Ghosh’s contribution to statistics: A brief outline

Clarke, Bertrand and Ghosal, Subhashis, Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 2008 - Orthogonal polynomial ensembles in probability
theory

König, Wolfgang, Probability Surveys, 2005 - Limiting shape for directed percolation models

Martin, James B., The Annals of Probability, 2004 - Oded Schramm: From circle packing to SLE

Rohde, Steffen, The Annals of Probability, 2011