Abstract
Recently the study of noise sensitivity and noise stability of Boolean functions has received considerable attention. The purpose of this paper is to extend these notions in a natural way to a different class of perturbations, namely those arising from running the symmetric exclusion process for a short amount of time. In this study, the case of monotone Boolean functions will turn out to be of particular interest. We show that for this class of functions, ordinary noise sensitivity and noise sensitivity with respect to the complete graph exclusion process are equivalent. We also show this equivalence with respect to stability. After obtaining these fairly general results, we study “exclusion sensitivity” of critical percolation in more detail with respect to medium-range dynamics. The exclusion dynamics, due to its conservative nature, is in some sense more physical than the classical i.i.d. dynamics. Interestingly, we will see that in order to obtain a precise understanding of the exclusion sensitivity of percolation, we will need to describe how typical spectral sets of percolation diffuse under the underlying exclusion process.
References
Benjamini I., Kalai G., Schramm O.: Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90, 5–43 (1999)
Diaconis P., Saloff-Coste L.: Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3(3), 696–730 (1993)
Garban C., Pete G., Schramm O.: The Fourier spectrum of critical percolation. Acta Math. 205(1), 19–104 (2010)
Garban, C., Steif, J.: Lectures on Noise Sensitivity and Percolation. arXiv:1102.5761 [math.PR]
Jara M., Landim C.: Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder. Ann. Inst. Henri Poincaré Probab. Stat. 44(2), 341–361 (2008)
Liggett T.M.: Interacting Particle Systems, pp. xvi+496. Springer, Berlin (2005)
Liggett T.M.: Stochastic Interacting Systems: Contact, Voter and Excusion Processes. Springer, Berlin (1999)
Morrow G.J., Zhang Y.: The sizes of the pioneering, lowest crossing and pivotal sites in critical percolation on the triangular lattice. Ann. Appl. Probab. 15, 1832–1886 (2005)
Mossel E., O’Donnell R.: On the noise sensitivity of monotone functions. Random Struct. Algorithms 23, 333–350 (2003)
Schramm O., Steif J.: Quantitative noise sensitivity and exceptional times for percolation. Ann. Math. 171, 619–672 (2010)
Smirnov S., Werner W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8(5–6), 729–744 (2001)
Werner, W.: Lectures on two-dimensional critical percolation. IAS Park City Graduate Summer School, arXiv:0710.0856 [math.PR] (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Broman, E.I., Garban, C. & Steif, J.E. Exclusion sensitivity of Boolean functions. Probab. Theory Relat. Fields 155, 621–663 (2013). https://doi.org/10.1007/s00440-011-0409-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-011-0409-9
Keywords
- Noise sensitivity
- Exclusion sensitivity
Mathematics Subject Classification (2010)
- 60K35
- 42B05