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Mean-field bounds for Poisson-Boolean percolation. (English) Zbl 1524.60264

Summary: We establish the mean-field bounds \(\gamma \ge 1\), \(\delta \ge 2\) and \(\Delta \ge 2\) on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously known only in the special case of fixed radii (in the case of \(\gamma \)), or not at all (in the case of \(\delta\) and \(\Delta \)). We deduce these as consequences of the mean-field bound \(\beta \le 1\), recently established under the same moment condition [H. Duminil-Copin et al., Ann. Henri Lebesgue 3, 677–700 (2020; Zbl 1454.82016)], using a relative entropy method introduced by the authors in previous work [“Upper bounds on the one-arm exponent for dependent percolation models”, Probab. Theory Relat. Fields 185, No. 1–2, 41–88 (2023; doi:10.1007/s00440-022-01176-3)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
60D05 Geometric probability and stochastic geometry

Citations:

Zbl 1454.82016
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References:

[1] D. Ahlberg, V. Tassion, and A. Teixeira, Sharpness of the phase transition for continuum percolation, Probab. Theory Related Fields 172 (2018), no. 1-2, 525-281. · Zbl 1404.60143
[2] M. Aizenman and D. J. Barsky, Sharpness of the phase transition in percolation models, Comm. Math. Phys. 108 (1987), no. 3, 489-526. · Zbl 0618.60098 · doi:10.1007/BF01212322
[3] M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys. 36 (1984), 107-143. · Zbl 0586.60096
[4] S. Chatterjee, J. Hanson, and P. Sosoe, Subcritical connectivity and some exact tail exponents in high dimensional percolation (2021). 2107.14347
[5] J. T. Chayes and L. Chayes, The mean field bound for the order parameter of Bernoulli percolation, Percolation Theory and Ergodic Theory of Infinite Particle Systems. The IMA Volumes in Mathematics and its Applications, vol. 8 (H. Kesten, ed.), Springer, New York, 1987, pp. 49-71. · Zbl 0621.60111
[6] A. Dembo and O. Zeitouni, Large deviations techniques and applications, Springer Berlin, Heidelberg, 2010. · Zbl 1177.60035
[7] V. Dewan and S. Muirhead, Upper bounds on the one-arm exponent for dependent percolation models, Probab. Theory Related Fields. 185 (2023), no. 1-2, 41-88.
[8] H. Duminil-Copin, A. Raoufi, and V. Tassion, Subcritical phase of d-dimensional Poisson-Boolean percolation and its vacant set, Ann. H. Lebesgue 3 (2020), 677-700. · Zbl 1454.82016
[9] R. Durrett and B. Nguyen, Thermodynamic inequalities for percolation, Commun. Math. Phys. 99 (1985), 253-269.
[10] R. Fitzner and R. van der Hofstad, Mean-field behavior for nearest-neighbor percolation in \[d\textgreater 10\], Electron. J. Probab. 22 (2017), 65 pp. · Zbl 1364.60130
[11] E. N. Gilbert, Random plane networks, J. Soc. Indust. Appl. Math. 9 (1961), 533-543. · Zbl 0112.09403
[12] J.-B. Gouréré, Subcritical regimes in the Poisson Boolean model of continuum percolation, Ann. Probab. 36 (2008), no. 4, 1209-1220. · Zbl 1148.60077
[13] G. R. Grimmett, Percolation, Springer, 1999.
[14] P. Hall, On continuum percolation, Ann. Probab. 13 (1985), no. 4, 1250-1266. · Zbl 0588.60096
[15] T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Comm. Math. Phys. 128 (1990), 333-391. · Zbl 0698.60100
[16] M. Heydenreich, R. van der Hofstad, G. Last, and K. Matzke, Lace expansion and mean-field behaviour for the random connection model (2019). 1908.11356
[17] T. Hutchcroft, Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters (2020). 2002.02916
[18] T. Hutchcroft, On the derivation of mean-field percolation critical exponents from the triangle condition, J. Stat. Phys. 189 (2022), no. 6. · Zbl 1495.82012
[19] T. Hutchcroft, E. Michta, and G. Slade, High-dimensional near-critical percolation and the torus plateau, Ann. Probab. (to appear). 2107.12971 · Zbl 1532.60216
[20] S. Kullback, Information theory and statistics, Dover, 1978.
[21] G. Last, M. D. Penrose, and S. Zuyev, On the capacity functional of the infinite cluster of a Boolean model, Ann. Appl. Probab. 27 (2017), no. 3, 1678-1701. · Zbl 1373.60160
[22] S. Lee, An inequality for greedy lattice animals, Ann. Appl. Probab. 3 (1993), no. 4, 1170-1188. · Zbl 0784.60049
[23] T. M. Liggett, R. H. Schonmann, and A. M. Stacey, Domination by product measures, Ann. Probab. 25 (1997), no. 1, 71-95. · Zbl 0882.60046
[24] R. Meester and R. Roy, Continuum percolation, Cambridge University Press, 2008.
[25] C. M. Newman, Some critical exponent inequalities for percolation, J. Stat. Phys. 45 (1986), 359-368.
[26] C. M. Newman, Another critical exponent inequality for percolation: \[ \beta \ge 2/ \delta \], J. Stat. Phys. 47 (1987), 695-699.
[27] M. D. Penrose, Random geometric graphs, Oxford University Press, 2003. · Zbl 1029.60007
[28] S. Ziesche, Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on \[{\mathbb{R}^d} \], Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 866-878 · Zbl 1391.60246
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