, [1/2, 2], one can find constant a = a(? ? ), b = b(? ? ) and M = M (? ? , c 0 ) as above that are uniform in ?. This ends the proof. As in the continuous case, we can deduce that the one-arm event decreases polynomially fast
, If 0 < r < s < +?, we write A(r, s) = [?s, s] 2 \] ? r, r[ 2 and we write Arm ? 0 (r, s (resp. Arm * ,? 0 (r, s)) for the event that there is an ?-black path rom the inner boundary of A(r, s) to its outer boundary made of black edges (resp. that lives in the white region of the plane) in the discrete percolation model of mesh ? defined in the beginning of Section 3
, Assume that f satisfies Conditions 1.7, 1.8, 1.10 as well as Condition 1.9 for some ? > 4. There exists C = C(?) < +? and ? = ?(?) > 0 such that, for each ? ?]0, 1], for each s ? [1, +?[ and r ? [1, s[: P [Arm ? 0 (r, s)] , P Arm * ,? 0 (r, s) ? C (r/s) ?
, First note that, since f and ?f have the same law, we have: 8 (P Arm ?, * 0 (r + ?, s ? ?) ?) P [Arm ? 0 (r, s)] ? P Arm ?
, With the same hypotheses as Proposition B.6, for each ? ?]0, 1] a.s. there is no unbounded black component in the discrete percolation model of mesh ? defined in the beginning of Section 3
, Boundedness of level lines for two-dimensional random fields. The Annals of Probability, pp.1653-1674, 1996.
, Random fields and geometry, 2007.
, Sharpness of the phase transition for continuum percolation in R 2, 2016.
, Level sets and extrema of random processes and fields, 2009.
, Percolation of random nodal lines. arXiv preprint arXiv:1605.08605, to appear in Inst, Hautes Etudes Sci. Publ. Math, 2016.
DOI : 10.1007/s10240-017-0093-0
URL : https://hal.archives-ouvertes.fr/hal-01321096
, Percolation without FKG. arXiv preprint, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01626674
, Discretisation schemes for level sets of planar Gaussian fields. arXiv preprint, 2017.
, Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials, 2017.
, The critical probability for random Voronoi percolation in the plane is 1/2. Probability theory and related fields, pp.417-468, 2006.
, Random wavefunctions and percolation, Journal of Physics A: Mathematical and Theoretical, vol.40, issue.47, p.14033, 2007.
DOI : 10.1088/1751-8113/40/47/001
URL : https://hal.archives-ouvertes.fr/hal-00186466
, Crossing probabilities for critical Bernoulli percolation on slabs. arXiv preprint, 2015.
, A course in approximation theory, American Mathematical Soc, vol.101, 2009.
DOI : 10.1090/gsm/101
, Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model, Communications on Pure and Applied Mathematics, vol.337, issue.2, pp.1165-1198, 2011.
DOI : 10.1016/j.crma.2003.08.003
URL : https://infoscience.epfl.ch/record/200293/files/Connection probabilities and RSWtype bounds for the twodimensional FK Ising model - 20370_ftp.pdf
, The box-crossing property for critical two-dimensional oriented percolation. arXiv preprint arXiv:1610, 2016.
DOI : 10.1007/s00440-017-0790-0
, Percolation (Grundlehren der mathematischen Wissenschaften ), 1999.
, Probability on graphs. Lecture Notes on Stochastic Processes on Graphs and Lattices, 2010.
DOI : 10.1017/cbo9780511762550
, Exponential rarefaction of real curves with many components, Publications math??matiques de l'IH??S, vol.1998, issue.1, pp.69-96, 2011.
DOI : 10.1155/S107379289800021X
URL : https://hal.archives-ouvertes.fr/hal-00484611
, A lower bound for the critical probability in a certain percolation process, Mathematical Proceedings of the Cambridge Philosophical Society, vol.21, issue.01, pp.13-20, 1960.
DOI : 10.1007/978-1-4684-9440-2
, Variance of the volume of random real algebraic submanifolds. arXiv preprint, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01354222
, Percolation in random fields. i. Theoretical and Mathematical Physics, pp.478-484, 1983.
, Percolation in random fields. ii, Theoretical and Mathematical Physics, vol.55, issue.3, pp.592-599, 1983.
, Percolation in random fields. iii, Theoretical and Mathematical Physics, vol.67, issue.2, pp.434-439, 1986.
, On the number of nodal domains of random spherical harmonics, American Journal of Mathematics, vol.131, issue.5, pp.1337-1357, 2009.
DOI : 10.1353/ajm.0.0070
, Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, 2015.
, Critical Percolation and the Minimal Spanning Tree in Slabs, Communications on Pure and Applied Mathematics, vol.44, issue.5, pp.2084-2120, 2017.
DOI : 10.1145/237814.237880
, Positively correlated normal variables are associated. The Annals of Probability, pp.496-499, 1982.
DOI : 10.1214/aop/1176993872
URL : http://doi.org/10.1214/aop/1176993872
, A note on percolation. Probability Theory and Related Fields, pp.39-48, 1978.
DOI : 10.1007/bf00535274
, The critical threshold for Bargmann- Fock percolation. arXiv preprint, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01634286
, Percolation probabilities on the square lattice, Annals of Discrete Mathematics, vol.3, pp.227-245, 1978.
, Crossing probabilities for Voronoi percolation. The Annals of Probability, pp.3385-3398, 2016.
DOI : 10.1214/15-aop1052
URL : http://arxiv.org/pdf/1410.6773
, Grenoble Alpes UMP5582, Institut Fourier, 38000 Grenoble, France alejandro.rivera@univ-grenoble-alpes.fr Supported by the ERC grant Liko No 676999
, 69100 Villeurbanne, France vanneuville@math.univ-lyon1.fr http://math.univ-lyon1.fr/ ~ vanneuville