×

zbMATH — the first resource for mathematics

Busemann functions and infinite geodesics in two-dimensional first-passage percolation. (English) Zbl 1293.82014
The first-passage percolation problem is studied on \(\mathbb{Z}^2\), addressing questions related to the existence and coalescence of infinite geodesics. The problem in turn is related to the structure of the set of ground states of the Edwards-Anderson model, see [L.-P. Arguin and M. Damron, Ann. Inst. Henri Poincaré, Probab. Stat. 50, No. 1, 28–62 (2014; Zbl 1292.82044)]. The departure point are papers by C. Newman and J. Wehr containing a rigorous study of infinite geodesics (and that of ground states of disordered ferromagnetic spin models), under a number of strong assumptions. The authors set a framework for working with distributional limits of Buseman functions and use them to prove some of Newman’s results under minimal assumptions. A purely directional condition is introduced which replaces Newman’s global curvature condition and implies the existence of directional geodesics. Interestingly, the existence of infinite geodesics which are directed in sectors can be proved without the latter condition. An analysis of distributional limits of geodesic graphs allows to prove the nonexistence of infinite backward paths. This is related to the almost-sure coalescence and a conjectured nonexistence of certain types of bigeodesics.

MSC:
82B43 Percolation
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alexander, K., Approximation of subadditive functions and convergence rates in limiting-shape results, Ann. Probab., 25, 30-55, (1997) · Zbl 0882.60090
[2] Arguin, L.-P., Damron, M.: On the number of ground states in the Edwards-Anderson spin glass model. To appear in Ann. Instit. Henri Poincaré (B) Probab. Statist, available at http://arxiv.org/abs/1110.6913v1 [math.PR], 2011 · Zbl 0783.60103
[3] Arguin, L.-P.; Damron, M.; Newman, C.; Stein, D., Uniqueness of ground states for short-range spin glasses in the half-plane, Commun. Math. Phys., 300, 641-657, (2010) · Zbl 1203.82101
[4] Auffinger, A. Damron. M.: Differentiability at the edge of the percolation cone and related results in first-passage percolation. Probab. Theor. Relat. Fields. 156(1-2), 193-227 (2013) · Zbl 1275.60093
[5] Benjamini, I.; Lyons, R.; Peres, Y., Group-invariant percolation on graphs, Geom. Funct. Anal., 9, 29-66, (1999) · Zbl 0924.43002
[6] Blair-Stahn, N.D.: First passage percolation and competition models. http://arxiv.org/abs/1005.0649v1 [math. PR], 2010 · Zbl 1067.60098
[7] Boivin, D., First passage percolation: the stationary case, Probab. Theor. Relat. Fields., 86, 491-499, (1990) · Zbl 0685.60103
[8] Burton, R.; Keane, M., Density and uniqueness in percolation, Commun. Math. Phys., 121, 501-505, (1989) · Zbl 0662.60113
[9] Coupier, D., Multiple geodesics with the same direction, Electron. Commun. Probab., 16, 517-527, (2011) · Zbl 1244.60093
[10] Cox, J.T.; Durrett, R., Some limit theorems for percolation processes with necessary and sufficient conditions, Ann. Probab., 4, 583-603, (1981) · Zbl 0462.60012
[11] Damron, M., Hochman, M.: Examples of non-polygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models. Ann. Appl. Probab. 23(3), 1074-1085 (2013) · Zbl 1314.60154
[12] Durrett, R.; Liggett, T., The shape of the limit set in richardson’s growth model, Ann. Probab., 9, 186-193, (1981) · Zbl 0457.60083
[13] Ferrari, P.A.; Pimentel, L.P.R., Competition interfaces and second class particles, Ann. Probab., 33, 1235-1254, (2006) · Zbl 1078.60083
[14] Forgacs G., Lipowsky, R., Nieuwenhuizen, T.M.: The behavior of interfaces in ordered and disordered systems. In: Domb, C., Lebowitz, J. (eds.) Phase transitions and critical phenomena, Vol. 14. London: Academic, 1991, pp. 135-363 · Zbl 1078.60083
[15] Grimmett, G., Kesten, H.: Percolation since Saint-Flour. http://arxiv.org/abs/1207.0373v1 [math.PR] (2012) · Zbl 1260.60007
[16] Garet, O.; Marchand, R., Coexistence in two-type first-passage percolation models, Ann. Appl. Probab., 15, 298-330, (2005) · Zbl 1080.60092
[17] Gouéré, J.-B., Shape of territories in some competing growth models, Ann. Appl. Probab., 17, 1273-1305, (2007) · Zbl 1132.60326
[18] Häggström, O., Infinite clusters in dependent automorphism invariant percolation on trees, Ann. Probab., 25, 1423-1436, (1997) · Zbl 0895.60098
[19] Häggström O.: Invariant percolation on trees and the mass-transport method. In: Bulletin of the International Statistical Institute. In: 52\^{nd} session proceedings, Tome LVIII, Book 1, Helsinki, 1999, pp. 363-366 · Zbl 0662.60113
[20] Häggström, O.; Meester, R., Asymptotic shapes for stationary first passage percolation, Ann. Probab., 23, 1511-1522, (1995) · Zbl 0852.60104
[21] Häggström, O.; Pemantle, R., First-passage percolation and a model for competing growth, J. Appl. Probab., 35, 683-692, (1998) · Zbl 0920.60085
[22] Hammersley, J.M., Welsh D.J.A.: First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In: Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif. New York: Springer, 1965, pp. 61-110 · Zbl 0143.40402
[23] Hoffman, C., Coexistence for Richardson type competing spatial growth models, Ann. Appl. Probab., 15, 739-747, (2005) · Zbl 1067.60098
[24] Hoffman, C., Geodesics in first-passage percolation, Ann. Appl. Probab., 18, 1944-1969, (2008) · Zbl 1153.60055
[25] Howard, C.D.; Newman, C.M., Geodesics and spanning trees for Euclidean first passage percolation, Ann. Probab., 29, 577-623, (2001) · Zbl 1062.60099
[26] Huse, D.A.; Henley, C.L., Pinning and roughening of domain walls in Ising systems due to random impurities, Phys. Rev. Lett., 54, 2708-2711, (1985)
[27] Kallenberg, O.: Foundations of modern probability, 2nd edn. Berlin: Springer, 2002 · Zbl 0996.60001
[28] Kesten, H., On the speed of convergence in first-passage percolation, Ann. Appl. Probab., 3, 296-338, (1993) · Zbl 0783.60103
[29] Krug, J., Spohn, H.: Kinetic roughening of growing surfaces. In: Godrécho, C. (ed.) Solids far from equilibrium: growth, morphology and defects. Cambridge: Cambridge University Press, 1991, pp. 479-582 · Zbl 0937.82020
[30] Licea, C.; Newman, C., Geodesics in two-dimensional first-passage percolation, Ann. Probab., 24, 399-410, (1996) · Zbl 0863.60097
[31] Newman, C.: A surface view of first-passage percolation. In: Proceedings of the international congress of mathematicians, Vols. 1, 2. (Zürich, 1994), Basel: Birkhäuser, 1995, pp. 1017-1023 · Zbl 0848.60089
[32] Newman, C.: Topics in disordered systems. In: Lectures in mathematics ETH Zürich. Basel: Birkhäuser Verlag, 1997 · Zbl 0835.60087
[33] Newman, C.; Piza, M., Divergence of shape fluctuations in two dimensions, Ann. Probab., 23, 977-1005, (1995) · Zbl 0835.60087
[34] Wehr, J., On the number of infinite geodesics and ground states in disordered systems, J. Statist. Phys., 87, 439-447, (1997) · Zbl 0937.82020
[35] Wehr, J.; Woo, J., Absence of geodesics in first-passage percolation on a half-plane, Ann. Probab., 26, 358-367, (1998) · Zbl 0937.60092
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.