Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T14:41:23.914Z Has data issue: false hasContentIssue false
  • English
  • Français

Simple peeling of planar maps with application to site percolation

Part of: Special processes Graph theory

Published online by Cambridge University Press:  26 February 2021

Timothy Budd  and
Nicolas Curien
Timothy Budd*
Affiliation:
Institute for Mathematics, Astrophysics and Particle Physics, Faculty of Science, Radboud University, Nijmegen, The Netherlands
Nicolas Curien
Affiliation:
Université Paris-Saclay, Orsay, France and Institut Universitaire de France, Paris, France e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolations on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations, distinct vertices on the exploration contour may, at latter stage, be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative “simple” peeling exploration of Angel and uncover deep relations with the lazy-peeling process. Along the way, we define and study the random Boltzmann map of the half-plane with a simple boundary for an arbitrary critical weight sequence. Its construction is nontrivial especially in the “dense regime,” where the half-planar random Boltzmann map does not possess an infinite simple core.

Keywords

Random planar mapspeeling processsimple boundaryBernoulli site percolation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 907 - 941
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first author acknowledges support of the START-UP 2018 programme with project number 740.018.017, which is financed by the Dutch Research Council (NWO). The second author is supported by ERC GeoBrown (ERC Advanced Grant 740943).

References

Abraham, R., Delmas, J.-F., and Guo, H., Critical multi-type Galton–Watson trees conditioned to be large. J. Theor. Probab. 31(2018), 757–788.CrossRefGoogle Scholar
Ambjørn, J., Budd, T., and Makeenko, Y., Generalized multi-critical one-matrix models. Nucl. Phys. B 913(2016), 357–380.CrossRefGoogle Scholar
Angel, O., Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13(2003), 935–974.CrossRefGoogle Scholar
Angel, O., Scaling of percolation on infinite planar maps, I. Preprint, 2005, arxiv:0501006 Google Scholar
Angel, O. and Curien, N., Percolations on random maps I: half-plane models. Ann. Inst. Henri Poincaré Probab. Stat. 51(2015), 405–431.CrossRefGoogle Scholar
Angel, O. and Ray, G., Classification of half planar maps . Ann. Probab. 43(2015), 1315–1349.CrossRefGoogle Scholar
Angel, O. and Ray, G., The half plane UIPT is recurrent. Probab. Theory Relat. Fields. 170(2018), 657–683.CrossRefGoogle ScholarPubMed
Angel, O. and Schramm, O., Uniform infinite planar triangulation. Commun. Math. Phys. 241(2003), 191–213.CrossRefGoogle Scholar
Baur, E., Miermont, G., and Richier, L., Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary. ALEA Lat. Amer. J. Probab. Math. Stat. 13(2016), 1123–1149.CrossRefGoogle Scholar
Benjamini, I. and Curien, N., Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points. Geom. Funct. Anal. 23(2013), 501–531.CrossRefGoogle Scholar
Bernardi, O., Curien, N., and Miermont, G., A Boltzmann approach to percolation on random triangulations. Can. J. Math. 71(2019), 1–43.CrossRefGoogle Scholar
Bernardi, O., Holden, N., and Sun, X., Percolation on triangulations: a bijective path to Liouville quantum gravity. Preprint, 2018. arxiv:1807.01684 Google Scholar
Bertoin, J., Curien, N., and Kortchemski, I., Random planar maps & growth-fragmentations. Ann. Probab. 46(2018), 207–260.CrossRefGoogle Scholar
Bertoin, J. and Doney, R. A., On conditioning a random walk to stay nonnegative. Ann. Probab. 22(1994), 2152–2167.CrossRefGoogle Scholar
Björnberg, J. E. and Stefánsson, S. Ö., On site percolation in random quadrangulations of the half-plane. J. Stat. Phys. 160(2015), 336–356.CrossRefGoogle Scholar
Bouttier, J. and Guitter, E., Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop. J. Phys. A Math. Theor. 42(2009), 465208.CrossRefGoogle Scholar
Brézin, E., Itykson, C., Parisi, G., and Zuber, J.-B., Planar diagrams. Commun. Math. Phys. 59(1978), 35–51.CrossRefGoogle Scholar
Budd, T., The peeling process of infinite Boltzmann planar maps . Electron. J. Comb. 23(2016), 1–28.Google Scholar
Budd, T. and Curien, N., Geometry of infinite planar maps with high degrees. Electron. J. Probab. 22(2017), 1–37.CrossRefGoogle Scholar
Budd, T., Curien, N., and Marzouk, C., Infinite random planar maps related to Cauchy processes. J. Éc. polytech. Math. 5(2019), 749–791.CrossRefGoogle Scholar
Caraceni, A. and Curien, N., Self-avoiding walks on the UIPQ . In: Sojourns in probability theory and statistical physics—III, Springer, Singapore, 2019, pp. 138–165.CrossRefGoogle Scholar
Caravenna, F. and Chaumont, L., Invariance principles for random walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probab. Stat. 44(2008), 170–190.CrossRefGoogle Scholar
Curien, N., A glimpse of the conformal structure of random planar maps. Commun. Math. Phys. 333(2015), 1417–1463.CrossRefGoogle Scholar
Curien, N., Planar stochastic hyperbolic triangulations. Probab. Theory Relat. Fields 165(2016), 509–540.CrossRefGoogle Scholar
Curien, N., Peeling random planar maps, Saint-Flour course. 2019. https://www.imo.universite-paris-saclay.fr/~curien/enseignement.html.Google Scholar
Curien, N. and Kortchemski, I., Percolation on random triangulations and stable looptrees. Probab. Theory Relat. Fields 163(2015), 303–337.CrossRefGoogle Scholar
Curien, N. and Miermont, G., Uniform infinite planar quadrangulations with a boundary. Random Structures Algorithms 47(2015), 30–58.CrossRefGoogle Scholar
Curien, N. and Richier, L., Duality of random planar maps via percolation. Ann. Inst. Fourier (2018), to appear.Google Scholar
Doney, R. A., The Martin boundary and ratio limit theorems for killed random walks. J. Lond. Math. Soc. (2) 58(1998), 761–768.CrossRefGoogle Scholar
Feller, W., An introduction to probability theory and its applications. Vol. II. 2nd ed., Wiley, New York, London, and Sydney, 1971.Google Scholar
Gwynne, E. and Miller, J., Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov–Hausdorff–Prokhorov–uniform topology. Electron. J. Probab. 22(2017), 1–47.CrossRefGoogle Scholar
Gwynne, E. and Miller, J., Metric gluing of Brownian and $\sqrt{8/3}$ -Liouville quantum gravity surfaces. Preprint, 2018. arxiv:1608.00955 CrossRefGoogle Scholar
Gwynne, E. and Miller, J., Convergence of the self-avoiding walk on random quadrangulations to SLE 8/3 on $\sqrt{8/3}$ -Liouville quantum gravity. Preprint, 2019. arxiv:1608.00956 Google Scholar
Gwynne, E. and Miller, J., Convergence of percolation on uniform quadrangulations with boundary to SLE 6 on $\sqrt{8/3}$ -Liouville quantum gravity. Preprint, 2021. arxiv:1701.05175 CrossRefGoogle Scholar
Holden, N. and Sun, X., Convergence of uniform triangulations under the Cardy embedding. Preprint, 2020. arxiv:1905.13207 Google Scholar
Ménard, L. and Nolin, P., Percolation on uniform infinite planar maps. Electron. J. Probab. 19(2014), 1–27.CrossRefGoogle Scholar
Neveu, J., Sur le théorème ergodique de Chung–Erdös. C. R. Acad. Sci. Paris Sér. I Math. 257(1963), 2953–2955.Google Scholar
Ray, G., Geometry and percolation on half planar triangulations. Electron. J. Probab. 19(2014), 1–28.CrossRefGoogle Scholar
Richier, L., Universal aspects of critical percolation on random half-planar maps. Electron. J. Probab. 20(2015), 1–45.CrossRefGoogle Scholar
Richier, L., The incipient infinite cluster of the uniform infinite half-planar triangulation. Electron. J. Probab. 23(2018), 1–38.CrossRefGoogle Scholar
Richier, L., Limits of the boundary of random planar maps. Probab. Theory Relat. Fields, 172(2018), 789–827.CrossRefGoogle Scholar
Sheffield, S., Quantum gravity and inventory accumulation. Ann. Probab. 44(2016), 3804–3848.CrossRefGoogle Scholar
Watabiki, Y., Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation . Nucl. Phys. B 441(1995), 119–163.CrossRefGoogle Scholar