Skip to Main Content


AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution


50 Years of First-Passage Percolation

About this Title

Antonio Auffinger, Northwestern University, Evanston, IL, Michael Damron, Georgia Institute of Technology, Atlanta, GA and Jack Hanson, The City College of New York, New York, NY

Publication: University Lecture Series
Publication Year: 2017; Volume 68
ISBNs: 978-1-4704-4183-8 (print); 978-1-4704-4356-6 (online)
DOI: https://doi.org/10.1090/ulect/068
MathSciNet review: MR3729447
MSC: Primary 60K35; Secondary 82B43

PDF View full volume as PDF

Read more about this volume

View other years and volumes:

Table of Contents

PDF Download chapters as PDF

Front/Back Matter

Chapters

References [Enhancements On Off] (What's this?)

References
  • Ahlberg, D. (2015). Asymptotics of first-passage percolation on one-dimensional graphs. Adv. in Appl. Probab., 47, 182–209.
  • Ahlberg, D. (2015). Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice. J. Theoret. Probab., 28, 198–222.
  • Ahlberg, D. (2015). A Hsu-Robbins-Erdős strong law in first-passage percolation. Ann. Probab., 43, 1992–2025.
  • Ahlberg, D., Damron, M. and Sidoravicius, V. (2016). Inhomogeneous first-passage percolation. Electron. J. Probab., 21, Paper no. 4, 19 pp.
  • Ahlberg, D. and Hoffman, C. (2016). Random coalescing geodesics in first-passage percolation. arXiv:1609.02447.
  • Aizenman, M. (1997). On the number of incipient spanning clusters. Nuclear Phys. B, 485, 551–582.
  • Alexander, K. (1997). Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab., 25, 30–55.
  • Alexander, K. (1993). A note on some rates of convergence in first-passage percolation. Ann. Appl. Probab., 3, no. 1, 81–90.
  • Alexander, K. (2011). Subgaussian rates of convergence of means in directed first passage percolation. arXiv: 1101.1549.
  • Alexander, K. and Zygouras, N. (2013). Subgaussian concentration and rates of convergence in directed polymers. Electron. J. Probab., 18, 1–28.
  • Alm, S. E. (1998). A note on a problem by Welsh in first-passage percolation. Combin. Probab. Comput., 7 (1), 11–15.
  • Alm, S. E. and Deijfen, M. (2015). First passage percolation on $\mathbb Z^{2}$ - a simulation study. J. Stat. Phys., 161, 657–678.
  • Alm, S. E. and Wierman, J. C. (1999). Inequalities for means of restricted first-passage times in percolation theory. Combin. Probab. Comput., 8, 307–315.
  • Anderson, G., Guionnet, A. and Zeitouni, O. (2010) An introduction to random matrices. Cambridge studies in advanced mathematics, 118.
  • Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab., 24, 1036–1048.
  • Antunović, T. and Procaccia, E. (2017). Stationary Eden model on Cayley graphs. Ann. Appl. Probab., 27, 517–549.
  • Auffinger, A. and Damron, M. (2013). The universal scaling relation for polymers in a random environment and related models ALEA, Lat. Am. J. Probab. Math. Stat., 10 (2), 857–880.
  • Auffinger, A. and Damron, M. (2014). A simplified proof of the relation between scaling exponents in first-passage percolation. Ann. Probab., 42, No. 3, 1197–1211.
  • Auffinger, A. and Damron, M. (2013). Differentiability at the edge of the limit shape and related results in first passage percoaltion. Probab. Theory Related Fields, 156, 193–227.
  • Auffinger, A., Damron, M. and Hanson, J. (2015). Limiting geodesics for first-passage percolation on subsets of $\Z ^2$, Ann. Appl. Probab., 25, 373–405.
  • Auffinger, A., Damron, M. and Hanson, J. (2014). Rate of convergence of the mean for sub-additive ergodic sequences. Adv. Math., 285, 138–181.
  • Auffinger, A. and Tang, S. (2016). On the time constant of high dimensional first passage percolation. Electron. J. Probab., 21, 23 pp.
  • Bakhtin, Y. and Wu, W. (2016). Transversal fluctuations for a first passage percolation model. arXiv: 1605.05965.
  • Basdevant, A. Enriquez, N., Gerin, L., Gouéré, J.-B. (2014). The shape of large balls in highly supercritical percolation. Electron. J. Probab., 19, 1–14.
  • Benaïm, M. and Rossignol, R. (2008). Exponential concentration for first passage percolation through modified Poincaré inequalities. : Ann. Inst. H. Poincaré Probab. Statist., 44, 544–573.
  • Benjamini, I. Kalai, G. and Schramm, O. (2003). First passage percolation has sublinear distance variance. Ann. Probab., 31, 1970–1978.
  • Benjamini, I. and Tessera, R. (2015). First passage percolation on nilpotent Cayley graphs and beyond. Electron. J. Probab., 20, no. 9, 20 pp.
  • van den Berg, J. (1983). A counterexample to a conjecture of J. M. Hammersley and D. J. A. Welsh concerning first-passage percolation. Adv. in Appl. Probab., 15, no. 2, 465–467.
  • van den Berg, J. and Fiebig, U. (1987). On a combinatorial conjecture concerning disjoint occurrences of events. Ann. Probab., 15, 354–375.
  • van den Berg, J. and Kesten, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab., 22, 556–569.
  • van den Berg, J. and Kesten, H. (1993). Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab., 3, 56–80.
  • van den Berg, J. and Kiss, D. (2012). Sublinearity of the travel-time variance for dependent first-passage percolation. Annals of Probab., 40, 743–764.
  • Bertrand, Q. and Pertinand, J. (2017). Dimension improvement in Dhar’s refutation of the Eden conjecture. arXiv:1705.08143.
  • Blair-Stahn, N. D. (2010). First passage percolation and competition models. arXiv:1005.0649.
  • Bjorklund, M. (2010). The asymptotic shape theorem for generalized first passage percolation. Ann. Probab., 38, 632–660.
  • Boivin, D. (1990). First passage percolation: the stationary case. Probab. Theory Related Fields, 86, 491–499.
  • Boivin, D. and Derrien, J.-M. (2002). Geodesics and recurrent of random walks in disordered systems. Ann. Probab., 7, 101–115.
  • Bollobás, B. (1979). Graph theory. Graduate Texts in Mathematics. Springer- Verlag, New York. An introductory course.
  • Boucheron, S, Lugosi, G., and Massart, P. Concentration inequalities. A nonasymptotic theory of independence. With a foreword by Michel Ledoux. Oxford University Press, Oxford, 2013. x+481 pp.
  • Bridson, M. and Haefliger, A. Metric Spaces of Non-Positive Curvature, Grundlder Math. Wiss. 319, Springer Verlag, 1999.
  • Bru, A., Albertos, S. Subiza, J., Garcia, J., and Bru, I. (2003). The Universal Dynamics of Tumor Growth. Biophys J., 85, 2948–2961.
  • Busemann, H. The geometry of geodesics. Vol. 6. Dover Publications, 1985.
  • Burago, D., Burago, Y. and Ivanov, S. A Course in Metric Geometry. American Mathematical Society, 2001.
  • Burkholder, D. L. (1973). Discussion of Prof. Kingman’s paper. Ann. Probab., 1, 900–902.
  • Burrows, E. M. (1991). Seaweeds of the British Isles. London: Natural History Museum. ISBN 0-565-00981-8.
  • Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys., 121, Number 3, 501–505.
  • Cerf, R. and Théret, M. (2014). Weak shape theorem in first passage percolation with infinite passage times. Ann. Inst. Henri Poincaré Probab. Stat., 52, no. 3, 1351–1381.
  • Cerf, R. and Théret, M. (2011). Law of the large numbers for the maximal flow through a domain of $\mathbb {R}^d$ in first passage percolation. Trans. Amer. Math. Soc., 363, 3665–3702.
  • Cerf, R. and Théret, M. (2011). Lower large deviations for the maximal flow through a domain of $\mathbb {R}^d$ in first passage percolation. Probab. Theory Related Fields, 150 no. 3, 635–661.
  • Cerf, R. and Théret, M. (2011). Upper large deviations for the maximal flow through a domain of $\mathbb {R}^d$ in first passage percolation. Ann. Appl. Probab., 21, No. 6, 2075–2108.
  • Chatterjee, S. (2013). The universal relation between scaling exponents in first-passage percolation. Ann. Math. (2), 177, no. 2, 663–697.
  • Chatterjee, S. and Dey, P. (2009). Central limit theorem for first-passage percolation time across thin cylinders. Probab. Theory Related Fields, 156, no. 3–4, 613–663.
  • Chattjerjee, S. and Dey, P. (2015). Multiple phase transitions in long-range first-passage percolation on square lattices. Comm. Pure Appl. Math., 69, No. 2, 203–256.
  • Chayes, L. (1991). On the critical behavior of the first passage time in $d \geq 3$. Helvetica Phys. Acta, 67, 1055–1071.
  • Chayes, J.T., Chayes, L. and Durrett, R. (1986). Critical behavior of the two-dimensional first passage time. J. Stat. Phys., 45, 933–951.
  • Chayes, J. T., Chayes, L. and Durrett, R. (1987). Inhomogeneous percolation problems and incipient infinite clusters. J. Phys. A, 20, 1521–1530.
  • Chayes, L. and Winfield, C. (1993). The density of interfaces: A new first passage problem. J. Appl. Prob., 30, 851–862.
  • Coupier, D. (2011). Multiple geodesics with the same direction. Electron. Commun. Probab., 16, 517–527.
  • Chow, Y. and Zhang, Y. (2003). Large deviations in first-passage percolation. Ann. Appl. Probab., 13, 1601–1614.
  • Couronné, O., Enriquez, N. and Gerin, L. (2011). Construction of a short path in high-dimensional first passage percolation. Electron. Commun. Probab., 16, 22–28.
  • Cox, J. T. (1980). The time constant of first-passage percolation on the square lattice. Adv. Appl. Prob., 12, 864–879.
  • Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation with necessary and sufficient conditions. Ann. Probab., 9, 583–603.
  • Cox, J. T. and Kesten, H. (1981). On the continuity of the time constant of first-passage percolation. J. Applied Prob., 18, 809–819.
  • Cranston, M. Gauthier, D. and Mountford, T. S. (2009). On large deviation regimes for random media models. Ann. Appl. Probab., 19, 826–862.
  • Damron, M., Hanson, J. and Sosoe, P. (2016). Subdiffusive concentration in first-passage percolation. Electron. J. Probab., 19, 27 pp.
  • Damron, M., Hanson, J. and Sosoe, P. (2015). Sublinear variance in first-passage percolation for general distributions. Probab. Theory Related Fields, 163, 223–258.
  • Damron, M. and Hochman, M. (2013). Examples of non-polygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models. Ann. Appl. Probab., 23, No. 3, 1074–1085.
  • Damron, M. and Hanson, J. (2014). Busemann functions and infinite geodesics in two-dimensional first-passage percolation. Comm. Math. Phys., 325, 917–963.
  • Damron, M. and Hanson, J. (2017). Bigeodesics in first-passage percolation. Comm. Math. Phys., 349, 753–776.
  • Damron, M. and Kubota, N. (2016). Rate of convergence in first-passage percolation under low moments. Stochastic Process. Appl., 126, no. 10, 3065–3076.
  • Damron, M., Lam, W.-K. and Wang, X. (2015). Asymptotics for $2D$ Critical First Passage Percolation. arXiv:1505.07544.
  • Damron, M. and Tang, P. (2016). Superlinearity of geodesic length in $2D$ critical first-passage percolation. arXiv:1610.02593.
  • Deijfen, M. and Häggström, O. (2006). The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model. Combin. Probab. Comput., 15, 345–353.
  • Deijfen, M. and Häggström, O. (2006). Nonmonotonic coexistence regions for the two- type Richardson model on graphs. Electron. J. Probab., 11, 331–344.
  • Deijfen, M. and Häggström, O. (2007). The two-type Richardson model with unbounded initial configurations. Ann. Appl. Probab., 17, 1639–1656.
  • Derrida, B. and Dickman, R. (1991). On the interface between two growing Eden clusters. Journal of Physics A: Mathematical and General, 24, 1385–1422.
  • Dhar, D. (1986). Asymptotic shape of Eden clusters, On growth and form. Ed. H. E. Stanley and N. Ostrowsky, Martinus Nijhoff., 288–292.
  • Duminil-Copin, H. and Tassion, V. (2015). A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb {Z}^d$. arXiv:1502.03051
  • Durrett, R. (2005). Probability: theory and examples. Duxbury Advanced Series. Third Edition.
  • Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Probab., 12, 999–1040.
  • Durrett, R. and Liggett, T. (1981). The shape of the limit set in Richardson’s growth model. Ann. Probab., 9, 186–193.
  • Eden, M. A two-dimensional growth process. 1961 Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. IV, 223–239, Univ. California Press, Berkeley, Calif.
  • Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist., 9, 586–596.
  • Falik, D. and Samorodnitsky, A. (2007). Edge-isoperimetric inequalities and influences. Combin. Probab. Comput., 16, 693–712.
  • Ferrari, P. A., Martin, J. B, and Pimentel, L. (2009). A phase transition for competition interfaces. Ann. Appl. Probab., 19, 281–317.
  • Ferrari, P. A. and Pimentel, L. (2005). Competition interfaces and second class particles. Ann. Probab., 33, 1235–1254.
  • Fishburn, P. C. and Shepp, L. A. (1991). On the FKB conjecture for disjoint intersections. Discrete Math., 98, 105–122.
  • Fontes, L. and Newman, C.M. (1993). First passage percolation for random colorings of $\Z ^d$. Ann. Appl. Probab., 3, 746–762.
  • Forgacs, G., Lipowsky R. and Nieuwenhuizen, T. (1991). The behavior of interfaces in ordered and disordered systems. In Phase transitions and critical phenomena (C. Domb and J. Lebowitz, eds.) 14 135–363. Academic, London.
  • Garet, O. (2009). Capacitive Flows on a 2D random net. Ann. Appl. Probab., 19, 641–660.
  • Garet, O. and Marchand, R. (2005). Coexistence in two-type first-passage percolation models. Ann. Appl. Probab., 15(1A), 298–330.
  • Garet, O. and Marchand, R. (2004). Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM Probab. Stat., 8, 169–199.
  • Garet, O. and Marchand, R. (2008). First-passage competition with different speeds: positive density for both species is impossible. Electron. J. Probab., 13, 2118–2159.
  • Garet, O. and Marchand, R. (2014). Large deviations for the contact process in random environment. Ann. Probab., 42, 1438–1479.
  • Garet, O., Marchand, R., Procaccia, E. and Théret, M. Continuity of the time and isoperimetric constants in supercritical percolation. Electron. Commun. Probab., 20, no. 92, 11 pp.
  • Garet, O., Gouere, J-B. and Marchand, R. (2013). The number of open paths in oriented percolation. arXiv:1312:2571.
  • Georgiou, N., Rassoul-Agha, F. and Seppäläinen, T. (2015). Geodesics and the competition interface for the corner growth model. arXiv:1510.00860.
  • Georgiou, N., Rassoul-Agha, F. and Seppäläinen, T. (2015). Stationary cocycles and Busemann functions for the corner growth model. arXiv:1510.00859.
  • Georgiou, N., Rassoul-Agha, F. and Seppäläinen, T. (2016). Variational formulas and cocycle solutions for directed polymer and percolation models. Comm. Math. Phys., 346, no. 2, 741–779.
  • Gouéré, J.-B. (2007). Shape of territories in some competing growth models. Ann. Appl. Probab., 17(4), 1273–1305.
  • Gouéré, J.-B. (2014). Monotonicity in first-passage percolation. ALEA Lat. Am. J. Probab. Math. Stat., 11, 565–569.
  • Grimmett, G. (1999). Percolation. 2nd edition. Springer, Berlin.
  • Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flow and electrical resistances Z..Wahrsch. Verw. Gebiete, 66, 335–366.
  • Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math., 97, 1061–1083.
  • Häggström, O. and Meester, R. (1995). Asymptotic shapes for stationary first passage percolation. Ann. Probab., 23, 1511–1522.
  • Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Probab., 35, 683–692.
  • Häggström, O. and Pemantle, R. (2000). Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model. Stochastic Process. Appl., 90, 207–222.
  • Hammersley, J. and Welsh, D. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. 1965 Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif., 61–110, Springer-Verlag, New York.
  • Hanson, J. (2013). Ph.D. thesis.
  • Higuchi, Y. and Zhang, Y. (2000). On the speed of convergence for two-dimensional first passage Ising percolation. Ann. Probab., 28, 353–378.
  • Hoffman, C. (2005). Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab., 15(1B), 739–747.
  • Hoffman, C. (2008). Geodesics in first passage percolation. Ann. Appl. Probab., 18, 1944–1969.
  • Howard, C. D. Models of first-passage percolation. Probability on discrete structures, 125–173, Encyclopaedia Math. Sci., 110, Springer, Berlin, 2004.
  • Howard, C. D. and Newman, C. M., From greedy lattice animals to Euclidean first-passage percolation. Perplexing Problems in Probability, (R. T. Durrett, ed.), Progr. Probab., 44, Birkhäuser, Boston, 107–119.
  • Howard, C. D. and Newman, C. M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab., 29, 577–623.
  • Huse, D. and Henley, C. (1985). Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett., 54, 2708–2711.
  • Huse, D. A., Henley, C. L., Fisher, D. S.: (1985). Huse, Henley, and Fisher respond. Phys. Rev. Lett., 55, 2924–2924.
  • Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys., 209, 437–476.
  • Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields, 116, 445–456.
  • Kardar, M. (1985). Roughening by impurities at finite temperatures. Phys. Rev. Lett., 55, 2923–2923.
  • Kardar, M., Parisi, G. and Zhang, Y. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett., 56, 889–892.
  • Kardar, M. and Zhang, Y.-C. (1987). Scaling of directed polymers in random media. Phys. Rev. Lett. , 56, 2087–2090.
  • Kesten, H. (1986). Aspects of first passage percolation. École d’Été de Probabilités de Saint Flour XIV, Lecture Notes in Mathematics, 1180, 125–264.
  • Kesten, H. (1987). Surfaces with minimal random weight and maximal flows: A higher dimensional version of first passage percolation, Illinois J. Math., 31, 99–166.
  • Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab., 3, 296–338.
  • Kesten, H. First-passage percolation. From classical to modern probability, 93–143, Progr. Probab., 54, Birkhäuser, Basel, 2003.
  • Kesten, H. and Zhang, Y. (1996). A central limit theorem for critical first-passage percolation in two dimensions. Probab. Theory Related Fields, 107, 137–160.
  • Kingman, J. F. C. (1968). The ergodic theory of sub additive stochastic processes. J. Roy. Statist. Soc. Ser., 30, 499–510.
  • Kingman, J. F. C. (1973). Subadditive ergodic theory. Ann. Probab., 1, 883–899.
  • Knuth, D. (1970). Permutations, matrices, and generalized Young tableaux. Pacific J. Math., 34, 709–727.
  • Krishnan, A. (2016). Variational formula for the time-constant of first-passage percolation I: Homogenization. Comm. Pure Appl. Math., 69, no. 10, 1984–2012.
  • Krug, J. (1987). Scaling relation for a growing surface. Phys. Rev. A, 36, 5465–5466.
  • Krug, J. and Spohn, H. Kinetic roughening of growing interfaces. Edited by C. Godrèche. Collection Aléa-Saclay: Monographs and Texts in Statistical Physics, 1. Cambridge University Press, Cambridge, 1992. xvi+588 pp. ISBN: 0-521-41170-X
  • Kubota, N. (2013). Rates of convergence in first passage percolation with low moment conditions. arXiv:1306.5917.
  • Kuhr, J.-T., Leisner, M. and Frey, E. (2011). Range expansion with mutation and selection: dynamical phase transition in a two-species Eden model. New Journal of Physics, 13.
  • Lagatta, T. and Wehr, J. (2010). A shape theorem for Riemannian first-passage percolation. J. Math. Phys., 51, 14 pp.
  • Lalley, S. (2003). Strict convexity of the limit shape in first-passage percolation. Electron. Commun. Probab., 8, 135–141.
  • Lee, S. (1993). An inequality for Greedy Lattice Animals. Ann. Appl. Probab., 3, 1170–1188.
  • Licea, C. and Newman, C. M. (1996). Geodesics in two-dimensional first-passage percolation. Ann. Probab., 24, 399–410.
  • Licea, C. and Newman, C. and Piza, M. (1996). Superdiffusivity in first-passage percolation. Probab. Theory Related Fields, 106, 559–591.
  • Lieb, E. H. and Loss, M. Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. xxii+346 pp. ISBN: 0-8218-2783-9.
  • Liggett, T. (1985). An Improved Subadditive Ergodic Theorem. Ann. Probab., 13, 4, 1279–1285.
  • Marchand, R. (2002). Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab., 12, 1001–1038.
  • Martin, J. (2004). Limiting shape for directed percolation models. Ann. Probab., 32, 2908–2937.
  • Mourrat, J.-C. (2012). Lyapunov exponents, shape theorems and large deviations for the random walk in random potential ALEA, Lat. Am. J. Probab. Math. Stat., 9, 165–209.
  • Nakajima, S. (2016). Maximal edge-traversal time in First Passage Percolation. arXiv:1605.04787
  • Newman, C. M. (1995). A surface view of first-passage percolation. Proc. International Congress of Math., Vol. 1, 2 (Zürich, 1994), 1017–1023, Birkhäuser, Basel.
  • Newman, C. M. Topics in disordered systems. Springer, 1997.
  • Newman, C. M. and Piza, M. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab., 23, 977–1005.
  • Pemantle, R. and Peres, Y. (1994). Planar first-passage percolation times are not tight. Probability and phase transition (Cambridge, 1993), 261-264, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 420, Kluwer Acad. Publ., Dordrecht.
  • Pimentel, L. P. R. (2007). Multitype shape theorems for first passage percolation models. Adv. Appl. Probab., 39(1), 53–76.
  • Reimer, D. (2000). Proof of the van den Berg-Kesten conjecture. Combin. Probab. Comput., 9, 27–32.
  • Rhee, W. (1995). On Rates of Convergence for Common Subsequences and First Passage Time. Ann. Appl. Probab., 5, 44–48.
  • Rockafellar, R. Convex Analysis. Princeton University Press, 1970.
  • Richardson, D. (1973). Random growth in a tessellation. Proc. Cambridge Philos. Soc., 74, 515–528.
  • Rossignol, R. (2006). Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Ann. Probab., 35, 1707–1725.
  • Rost, H. (1981). Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete, 58, 41–53.
  • Rossignol, R. and Théret. (2010). Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation. Stochastic Process. Appl., 120, 873–900.
  • Rossignol, R. and Théret. (2013). Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation. ESAIM Probab. Stat., 17, 70–104.
  • Smythe, R. and Wierman, J. C. (1977). First-passage percolation on the square lattice. I. Advances in Appl. Probability, 9, no. 1, 38–54.
  • Smythe, R. and Wierman, J. C. (1978). First-passage percolation on the square lattice. III. Advances in Appl. Probability, 10, no. 1, 155–171.
  • Sodin, S. (2014). Positive temperature versions of two theorems on first-passage percolation. Geometric Aspects of Functional Analysis, 2116, Lecture Notes in Mathematics, 441–453.
  • Steele, J. (1986). An Efron-Stein inequality for nonsymmetric statistics. Ann. Statist., 14, 753–758.
  • Steele, J. and Zhang, Y. (2003) Nondifferentiability of the time constants of first-passage percolation. Ann. Probab., 31, 1028–1051.
  • Talagrand, M. (1994). On Russo’s approximate zero-one law. Ann. Probab., 22, 1576–1587.
  • Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. I.H.E.S., 81, 73–205.
  • Tessera, R. (2014). Speed of convergence in first passage percolation and geodesicity of the average distance. arXiv:1410.1701.
  • Timár, A. (2013). Boundary-connectivity via graph theory. Proc. Amer. Math. Soc, 141, 475–480.
  • Tracy, C. A.; Widom, H. (1994). Level-spacing distributions and the Airy kernel, Comm. Math. Phys., 159, 151–174
  • Vahidi-Asl, M. Q. and Wierman, J. C. (1990). First-passage percolation on the Voronoi tessellation and Delaunay triangulation. Random Graphs ’87 , 341–359, Wiley, New York.
  • Vahidi-Asl, M. Q. and Wierman, J. C. (1992). A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation. Random Graphs ’87 , 247–262, Wiley, New York.
  • Van der Hofstad, R. (2014). Random Graphs and Complex Networks. Vol. II. Online at http://www.win.tue.nl/~rhofstad/NotesRGCNII.pdf.
  • Wehr, J. (1997). On the number of infinite geodesics and ground states in disordered systems. J. Statist. Phys., 87, 439–447.
  • Wehr, J. and Aizenman, M. (1987). Fluctuations of extensive functions of quenched random couplings. J. Stat. Phys., 60, 287–306.
  • Wehr, J. and Woo, J. (1998). Absence of geodesics in first-passage percolation on a half-plane. Ann. Probab., 26, 358–367.
  • Wierman, J. C., (1980). Weak moment conditions for time coordinates in first-passage percolation models. J. Appl. Probab., 17, 968–978.
  • Wierman, J. C. and Reh, W. (1978). On conjectures in first passage percolation theory. Ann. Probab., 6, 388–397.
  • Wolf, D. E. and Kertesz, J. (1987). Noise reduction in Eden models: I. J. Phys. A, 20, L257-L261.
  • Wüthrich, M. V. (1998). Scaling identity for crossing Brownian motion in a Possonian potential. Probab. Theory Related Fields, 112, 299–319.
  • Yao, C.-L. (2015). Limit theorems for critical first-passage percolation on the triangular lattice. arXiv: 1602.00065.
  • Zabolitsky, J. G. and Stauffer, D. (1986). Simulation of large Eden clusters. Phys. Rev. A 34 1523–1530.
  • Zhang, Y. (1993). A shape theorem for epidemics and forest fires with finite range interactions. Ann. Probab., 21, 1755–1781.
  • Zhang, Y. (2000). Critical behavior for maximal flows on the cubic lattice. J. Stat. Phys. 98, 799–811.
  • Zhang, Y. (2006). The divergence of fluctuations for shape in first passage percolation. Probab. Theory Related Fields, 136, 298–320.
  • Zhang, Y. (1999). Double behavior of critical first-passage percolation. Perplexing Problems in Probability, 143–158.
  • Zhang, Y. (2016). Limit theorems for maximum flows on a lattice. To appear in Probab. Theory Related Fields.
  • Zhang, Y. (2010). On the concentration and the convergence rate with a moment condition in first-passage percolation. Stochastic Process. Appl., 120, 1317–1341.
  • Zhang, Y. (2008). Shape fluctuations are different in different directions. Ann. Probab., 36, 331–362.
  • Zhang, Y. (1995). Supercritical behaviors in first-passage percolation. Stochastic Process. Appl., 59, 251–266.
  • Zhang, Y. and Zhang, Y. (1984). A limit theorem for $N_{0,n}/n$ in first-passage percolation. Ann. Probab., 12, 1068–1076.