Geodesics and spanning trees for Euclidean first-passage percolation

Annals of Probability

Volumn 29, Issue 2, 2001, Pages 577-623

  Howard, C Douglas ^a   Newman, Charles M ^b  

^a CITY UNIVERSITY OF NEW YORK   (United States)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

Combinatorial optimization; First passage percolation; Geodesic; Minimal spanning tree; Poisson process; Random metric; Random surface; Shape theorem

Indexed keywords

EID: 0035537282     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1008956686     Document Type: Article

References (48)

1. AIZENMAN, M. (1996). The geometry of critical percolation and conformal invariance. In The Nineteenth IUPAP International Conference on Statistical Physics (H. Bai-lin, ed.) 104-120. World Scientific, Singapore.
2. AIZENMAN, M. and BURCHARD, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419-453.
3. AIZENMAN, M., BURCHARD, A., NEWMAN, C. M. and WILSON, D. (1999). Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms 15 319-367.
4. ALDOUS, D. and STEELE, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247-258.
5. ALEXANDER, K. S. (1993). A note on some rates of convergence in first-passage percolation. Ann. Appl. Probab. 3 81-90.
6. ALEXANDER, K. S. (1995). Percolation and minimal spanning trees in infinite graphs. Ann. Probab. 23 87-104.
7. ALEXANDER, K. S. (1997). Approximations of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 30-55.
8. ALEXANDER, K. S. and MOLCHANOV, S. A. (1994). Percolation of level sets for two-dimensional random fields with lattice symmetry. J. Statist. Phys. 77 627-643.
9. BAIK, J. DEIFT, P. and JOHANSSON, K. (1999). On the distribution of the longest increasing subsequence in a random permutation. J. Amer. Math. Soc. 12 1119-1178.
10. BOIVIN, D. (1990). First-passage percolation: the stationary case. Probab. Theory Related Fields 86 491-499.
11. COX, J. T. and DURRETT, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583-603.
12. COX, J. T., GANDOLFI, A., GRIFFIN, P. S. and KESTEN, H. (1983). Greedy lattice animals I: upper bounds. Ann. App. Probab. 3 1151-1169.
13. DURRETT, R. (1991). Probability: Theory and Examples. Wadsworth, Pacific Grove, CA.
14. GANDOLFI, A. and KESTEN, H. (1994). Greedy lattice animals II: linear growth. Ann. Appl. Probab. 4 76-107.
15. GRIMMETT, G. (1989). Percolation. Springer, Berlin.
16. HAMMERSLEY, J. M. and WELSH, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In Bernoulli, Bayes, Laplace Anniversary Volume (J. Neyman and L. Le Cam, eds.) 61-110. Springer, Berlin.
17. HOWARD, C. D. (1998). Good paths don't double back. Amer. Math. Monthly 105 354-357.
18. HOWARD, C. D. and NEWMAN, C. M. (1997). Euclidean models of first-passage percolation. Probab. Theory Related Fields 108 153-170.
19. HOWARD, C. D. and NEWMAN, C. M. (1999). From greedy lattice animals to Euclidean first-passage percolation. In Perplexing Problems in Probability (M. Bramson and R. Durrett, eds.) 107-119. Birkhäuser Boston.
20. HUSE, D. A. and HENLEY, C. L. (1985). Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett. 54 2708-2711.
21. HUSE, D. A., HENLEY, C. L. and FISHER, D. S. (1985). Phys. Rev. Lett. 55 2924-2924.
22. JOHANSSON, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445-456.
23. KARDAR, M. (1985). Roughening by impurities at finite temperatures. Phys. Rev. Lett. 55 2923-2923.
24. KARDAR, M., PARISI, G. and ZHANG, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889-892.
25. KESTEN, H. (1986). Aspects of first-passage percolation. École d'Été de Probabilités de Saint-Flour XIV. Lecture Notes in Math. 1180 125-264. Springer, Berlin.
26. KESTEN, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296-338.
27. KRUG, J. and SPOHN, H. (1991). Kinetic roughening of growing surfaces. In Solids Far from Equilibrium: Growth, Morphology and Defects (C. Godrèche, ed.). Cambridge Univ. Press.
28. LICEA, C. and NEWMAN, C. M. (1996). Geodesics in two-dimensional first-passage percolation. Ann. Probab. 24 399-410.
29. MEESTER, R. and ROY, R. (1996). Continuum Percolation. Cambridge Univ. Press.
30. NAGAEV, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745-789.
31. NEVEU, J. (1972). Martingales á Temps Discret. Masson & Cie, Paris. [English translation by T. P. Speed (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.] Paris.
32. NEVEU, J. (1972). Martingales á Temps Discret. Masson & Cie, Paris. [English translation by T. P. Speed (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.] Paris.
33. NEWMAN, C. M. (1995). A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians (S. D. Chatterji, ed.) 1017-1023. Birkhäuser, Basel.
34. NEWMAN, C. M. (1997). Topics in Disordered Systems. Birkhäuser Basel.
35. NEWMAN, C. M. and PIZA, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977-1005.
36. NEWMAN, C. M. and STEIN, D. L. (1994). Spin glass model with dimension-dependent ground state multiplicity. Phys. Rev. Lett. 72 2286-2289.
37. NEWMAN, C. M. and STEIN, D. L. (1996). Ground state structure in a highly disordered spin-glass model. J. Statist. Phys. 92 1113-1132.
38. RICHARDSON, D. (1973). Random growth in a tesselation. Proc. Cambridge Philos. Soc. 74 515-528.
39. SCHRAMM, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288.
40. SERAFINI, H. C. (1997). First-passage percolation in the Delaunay graph of a d-dimensional Poisson process. Ph.D. dissertation, Courant Inst. of Math. Sciences, New York Univ.
41. SMYTHE, R. T. and WIERMAN, J. C. (1978). First-Passage Percolation on the Square Lattice. Lecture Notes in Math. 671 Springer, Berlin.
42. VAHIDI-ASL, M. Q. and WIERMAN, J. C. (1990). First-passage percolation on the Voronoi tessellation and Delaunay triangulation. In Random Graphs '87 (M. Karoński, J. Jaworski and A. Ruciński, eds.) 341-359. Wiley, New York.
43. VAHIDI-ASL, M. Q. and WIERMAN, J. C. (1992). A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation. In Random Graphs '89 (A. Frieze and T. Luczak, eds.) 247-262. Wiley, New York.
44. VAN DEN BERG, J. and KESTEN, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556-569.
45. WEHR, J. (1997). On the number of infinite geodesics and ground states in disordered systems. J. Statist. Phys. 87 439-447.
46. YUKICH, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, Berlin.
47. ZUEV, S. A. and SIDORENKO, A. F. (1985). Continuous models of percolation theory I. Theoret. and Math. Phys. 62 76-86 (51-58 in translation from Russian).
48. ZUEV, S. A. and SIDORENKO, A. F. (1985). Continuous models of percolation theory II. Theoret. and Math. Phys. 62 253-262 (171-177 in translation from Russian).