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Non-existence of bi-infinite geodesics in the exponential corner growth model

Part of:Special processes

Published online by Cambridge University Press:  16 November 2020

Márton Balázs,
Ofer Busani and
Timo Seppäläinen
Márton Balázs
Affiliation:
University of Bristol, School of Mathematics, Fry Building, Woodland Rd., Bristol BS8 1UG, UK, E-mail: m.balazs@bristol.ac.uk; https://people.maths.bris.ac.uk/~mb13434/
Ofer Busani
Affiliation:
University of Bristol, School of Mathematics, Fry Building, Woodland Rd., Bristol BS8 1UG, UK, E-mail: o.busani@bristol.ac.uk; https://people.maths.bris.ac.uk/~di18476/
Timo Seppäläinen
Affiliation:
University of Wisconsin-Madison, Mathematics Department, Van Vleck Hall, 480 Lincoln Dr., Madison, WI 53706-1388, USA, E-mail: seppalai@math.wisc.edu; http://www.math.wisc.edu/~seppalai

Abstract

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This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.

Keywords

bi-infinitecorner growth modeldirected percolationgeodesicrandom growth modellast-passage percolationqueues

MSC classification

Primary:60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary:60K37: Processes in random environments

Information

Type
Probability
Information
Forum of Mathematics, Sigma , Volume 8 , 2020, e46
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Ahlberg, Daniel and Hoffman, Christopher, ‘Random coalescing geodesics in first-passage percolation’, arXiv:1609.02447, 2016.Google Scholar
Aldous, David and Diaconis, Persi, ‘Hammersley’s interacting particle process and longest increasing subsequences’, Probab. Theory Related Fields, 103(2):199–213, 1995.CrossRefGoogle Scholar
Auffinger, Antonio, Damron, Michael, and Hanson, Jack, ‘Limiting geodesics for first-passage percolation on subsets of ${\mathbb{Z}}^2$ ’, Ann. Appl. Probab., 25(1):373–405, 2015.CrossRefGoogle Scholar
Auffinger, Antonio, Damron, Michael, and Hanson, Jack, 50 years of first-passage percolation , volume 68 of University Lecture Series, (American Mathematical Society, Providence, RI, 2017).Google Scholar
Balázs, Márton, Cator, Eric, and Seppäläinen, Timo, ‘Cube root fluctuations for the corner growth model associated to the exclusion process’, Electron. J. Probab., 11(42): 1094–1132 (electronic), 2006.CrossRefGoogle Scholar
Balázs, Márton, Komjáthy, Júlia, and Seppäläinen, Timo, ‘Microscopic concavity and fluctuation bounds in a class of deposition processes’, Ann. Inst. Henri Poincaré Probab. Stat., 48(1):151–187, 2012.CrossRefGoogle Scholar
Basu, Riddhipratim, Hoffman, Christopher, and Sly, Allan, ‘Nonexistence of bigeodesics in integrable models of last passage percolation’, arXiv:1811.04908, 2018.Google Scholar
Basu, Riddhipratim, Sidoravicius, Vladas, and Sly, Allan, ‘Last passage percolation with a defect line and the solution of the slow bond problem’, arXiv:1408.3464, 2014.Google Scholar
Cator, Eric and Groeneboom, Piet, ‘Second class particles and cube root asymptotics for Hammersley’s process’, Ann. Probab., 34(4):1273–1295, 2006.CrossRefGoogle Scholar
Chaumont, Hans and Noack, Christian, ‘Characterizing stationary $1+1$ dimensional lattice polymer models’, Electron. J. Probab., 23:Paper 38, 19, 2018.Google Scholar
Damron, Michael and Hanson, Jack, ‘Busemann functions and infinite geodesics in two-dimensional first-passage percolation’, Comm. Math. Phys., 325(3):917–963, 2014.CrossRefGoogle Scholar
Damron, Michael and Hanson, Jack, ‘Bigeodesics in first-passage percolation’, Comm. Math. Phys., 349(2):753–776, 2017.CrossRefGoogle Scholar
Dauvergne, Duncan, Ortmann, Janosch, and Virág, Bálint, ‘The directed landscape’, arXiv:1812.00309, 2018.Google Scholar
Eden, Murray, ‘A two-dimensional growth process’, In Proc. 4th Berkeley Sympos. Math. Statist. and Prob. , Vol. IV (Univ. California Press, Berkeley, Calif., 1961), 223–239.Google Scholar
Wai-Tong (Louis) Fan and Seppäläinen, Timo, ‘Joint distribution of Busemann functions in the exactly solvable corner growth model’, Journal Probability and Mathematical Physics, arXiv:1808.09069, 2018.Google Scholar
Feller, William, An introduction to probability theory and its applications , Vol. II ., 2e (John Wiley & Sons Inc., New York, 1971).Google Scholar
Georgiou, Nicos, Rassoul-Agha, Firas, and Seppäläinen, Timo, ‘Geodesics and the competition interface for the corner growth model’, Probab. Theory Related Fields, 169(1-2):223–255, 2017.CrossRefGoogle Scholar
Georgiou, Nicos, Rassoul-Agha, Firas, and Seppäläinen, Timo, ‘Stationary cocycles and Busemann functions for the corner growth model’, Probab. Theory Related Fields, 169(1-2):177–222, 2017.CrossRefGoogle Scholar
Gravner, Janko, Tracy, Craig A., and Widom, Harold, ‘Limit theorems for height fluctuations in a class of discrete space and time growth models’, J. Statist. Phys., 102(5-6):1085–1132, 2001.CrossRefGoogle Scholar
Hammersley, John M. and Welsh, Dominic J. A., ‘First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory’, In Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif , (Springer-Verlag, New York, 1965), 61–110.Google Scholar
Janjigian, Christopher and Rassoul-Agha, Firas, ‘Busemann functions and Gibbs measures in directed polymer models on ${\mathbb{Z}}^2$ ’, Ann. Probab., 48(2):778–816, 2020.CrossRefGoogle Scholar
Janjigian, Christopher, Rassoul-Agha, Firas, and Seppäläinen, Timo, ‘Geometry of geodesics through Busemann measures in directed last-passage percolation’, arXiv:1908.09040, 2019.Google Scholar
Johansson, Kurt, ‘Shape fluctuations and random matrices’, Comm. Math. Phys., 209(2):437–476, 2000.CrossRefGoogle Scholar
Johansson, Kurt, ‘Discrete orthogonal polynomial ensembles and the Plancherel measure’, Ann. of Math. (2), 153(1):259–296, 2001.CrossRefGoogle Scholar
Kesten, Harry, ‘Aspects of first passage percolation’, In École d’été de probabilités de Saint-Flour , XIV—1984, volume 1180 of Lecture Notes in Math . (Springer, Berlin, 1986), 125–264.Google Scholar
Licea, Cristina and Newman, Charles M., ‘Geodesics in two-dimensional first-passage percolation’, Ann. Probab., 24(1):399–410, 1996.CrossRefGoogle Scholar
Matetski, Konstantin, Quastel, Jeremy, and Remenik, Daniel, ‘The KPZ fixed point’, arXiv preprint arXiv:1701.00018, 2016.Google Scholar
Newman, Charles M., ‘A surface view of first-passage percolation’, In Proceedings of the International Congress of Mathematicians , Vol. 1, 2 (Zürich, 1994) (Birkhäuser, Basel, 1995), 1017–1023.CrossRefGoogle Scholar
Newman, Charles M., Topics in Disordered Systems, Lectures in Mathematics ETH Zürich, (Birkhäuser Verlag, Basel, 1997).CrossRefGoogle Scholar
O’Connell, Neil and Yor, Marc, ‘Brownian analogues of Burke’s theorem’, Stochastic Process. Appl., 96(2):285–304, 2001.CrossRefGoogle Scholar
Pimentel, Leandro P. R., ‘Duality between coalescence times and exit points in last-passage percolation models’, Ann. Probab., 44(5):3187–3206, 2016.CrossRefGoogle Scholar
Resnick, Sidney, Adventures in Stochastic Processes, (Birkhäuser Boston Inc., Boston, MA, 1992).Google Scholar
Seppäläinen, Timo, ‘Increasing sequences of independent points on the planar lattice’, Ann. Appl. Probab., 7(4):886–898, 1997.CrossRefGoogle Scholar
Seppäläinen, Timo, ‘Exact limiting shape for a simplified model of first-passage percolation on the plane’, Ann. Probab., 26(3):1232–1250, 1998.CrossRefGoogle Scholar
Seppäläinen, Timo, ‘The corner growth model with exponential weights’, In Random Growth Models, volume 75 of Proc. Sympos. Appl. Math. (Amer. Math. Soc., Providence, RI, 2018), 133–201, arXiv:1709.05771.CrossRefGoogle Scholar
Wehr, Jan and Woo, Jung, ‘Absence of geodesics in first-passage percolation on a half-plane’, Ann. Probab., 26(1):358–367, 1998.CrossRefGoogle Scholar