Abstract
The orthant model is a directed percolation model on \(\mathbb {Z}^{d}\), in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.
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Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Comm. Math Phys. 108(3), 489–526 (1987)
Beekenkamp, T.: Sharp phase transitions in percolation models. In preparation (2021)
Beekenkamp, T.: Sharpness of the phase transition for the corrupted compass model on transitive graphs. Indagationes Mathematicae New Series 32 (3), 736–744 (2021)
Dereudre, D., Houdebert, P.: Sharp phase transition for the continuum Widom-Rowlinson model Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 57(1), 387–407 (2021)
Duminil-Copin, H., Goswami, S., Rodriguez, P.-F., Severo, F.: Equality of critical parameters for percolation of Gaussian free field level-sets. arXiv:2002.07735 (2020)
Duminil-Copin, H., Raoufi, A., Tassion, V.: Exponential decay of connection probabilities for subcritical Voronoi percolation in \(\mathbb {R}^{d}\). Probab. Theory Relat. Fields, 1–12 (2017)
Duminil-Copin, H., Raoufi, A., Tassion, V.: Sharp phase transition for the random-cluster and Potts models via decision trees. Ann. Math. 189(1), 75–99 (2019)
Duminil-Copin, H., Raoufi, A., Tassion, V.: Subcritical phase of d-dimensional poisson–boolean percolation and its vacant set. Annales Henri Lebesgue 3, 677–700 (2020)
Grimmett, G.: Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], second edition. Springer, Berlin (1999)
Holmes, M., Salisbury, T.S.: Degenerate random environments. Random Structures & Algorithms 45(1), 111–137 (2014)
Holmes, M., Salisbury, T.S.: Random walks in degenerate random environments. Can. J. Math. 66(5), 1050–1077 (2014)
Holmes, M., Salisbury, T.S.: Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment. Electron. J. Probab. 22, 1–18 (2017)
Holmes, M., Salisbury, T.S.: Phase transitions for degenerate random environments. ALEA 18, 707–725 (2021)
Holmes, M., Salisbury, T.S.: A shape theorem for the orthant model. The Annals of Probability 49(3), 1237–1256 (2021)
Hutchcroft, T.: New critical exponent inequalities for percolation and the random cluster model. Probability and Mathematical Physics 1(1), 147–165 (2020)
Menshikov, M.V.: Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR 288(6), 1308–1311 (1986)
Muirhead, S., Vanneuville, H.: The sharp phase transition for level set percolation of smooth planar Gaussian fields. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 56(2), 1358–1390 (2020)
O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, Cambridge (2014)
O’Donnell, R., Saks, M., Schramm, O., Servedio, R.A.: Every decision tree has an influential variable. In: 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp 31–39. IEEE (2005)
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I thank Matija Pasch for insightful discussions on the topic, as well as for useful comments on the manuscript.
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Beekenkamp, T. Sharpness of the Phase Transition for the Orthant Model. Math Phys Anal Geom 24, 36 (2021). https://doi.org/10.1007/s11040-021-09408-z
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DOI: https://doi.org/10.1007/s11040-021-09408-z