## Limiting shape for directed percolation models.(English)Zbl 1065.60149

The author considers a class of first- and last-passage percolation models on orthants. Under rather weak moment conditions a shape theorem is proved, and continuity of the shape also at the boundaries, as well as the asymptotics in $$d= 2$$ at the boundary are found. Also the possibilities are discussed when the moment conditions of the theorem are not fulfilled, supported by numerical simulations.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C43 Time-dependent percolation in statistical mechanics 82B43 Percolation 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
Full Text:

### References:

 [1] Baccelli, F., Borovkov, A. and Mairesse, J. (2000). Asymptotic results on infinite tandem queueing networks. Probab. Theory Related Fields 118 365–405. · Zbl 0976.60088 [2] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178. JSTOR: · Zbl 0932.05001 [3] Baryshnikov, Y. (2001). GUEs and queues. Probab. Theory Related Fields 119 256–274. · Zbl 0980.60042 [4] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583–603. JSTOR: · Zbl 0462.60012 [5] Cox, J. T., Gandolfi, A., Griffin, P. S. and Kesten, H. (1993). Greedy lattice animals I: Upper bounds. Ann. Appl. Probab. 3 1151–1169. · Zbl 0818.60039 [6] Dembo, A., Gandolfi, A. and Kesten, H. (2001). Greedy lattice animals: Negative values and unconstrained maxima. Ann. Probab. 29 205–241. · Zbl 1016.60048 [7] Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation . Wadsworth, Belmont, CA. · Zbl 0659.60129 [8] Durrett, R. and Liggett, T. M. (1981). The shape of the limit set in Richardson’s growth model. Ann. Probab. 9 186–193. JSTOR: · Zbl 0457.60083 [9] Gandolfi, A. and Kesten, H. (1994). Greedy lattice animals II: Linear growth. Ann. Appl. Probab. 4 76–107. JSTOR: · Zbl 0824.60100 [10] Glynn, P. W. and Whitt, W. (1991). Departures from many queues in series. Ann. Appl. Probab. 1 546–572. JSTOR: · Zbl 0749.60090 [11] Gravner, J., Tracy, C. A. and Widom, H. (2001). Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102 1085–1132. · Zbl 0989.82030 [12] Hambly, B. M., Martin, J. B. and O’Connell, N. (2002). Concentration results for a Brownian directed percolation problem. Stochastic Process. Appl. 102 207–220. · Zbl 1075.60562 [13] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476. · Zbl 0969.15008 [14] 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. · Zbl 0602.60098 [15] Kesten, H. (1987). Percolation theory and first-passage percolation. Ann. Probab. 15 1231–1271. JSTOR: · Zbl 0629.60103 [16] Kesten, H. (1996). On the non-convexity of the time constant in first-passage percolation. Electron. Comm. Probab. 1 1–6. · Zbl 0866.60087 [17] Kesten, H. and Su, Z.-G. (2000). Asymptotic behavior of the critical probability for $$\rho$$-percolation in high dimensions. Probab. Theory Related Fields 117 419–447. · Zbl 0961.60093 [18] Marchand, R. (2002). Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12 1001–1038. · Zbl 1062.60100 [19] Martin, J. B. (2002). Large tandem queueing networks with blocking. Queueing Syst. Theory Appl. 41 45–72. · Zbl 1053.60098 [20] Martin, J. B. (2002). Linear growth for greedy lattice animals. Stochastic Process. Appl. 98 43–66. · Zbl 1060.60045 [21] Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977–1005. JSTOR: · Zbl 0835.60087 [22] O’Connell, N. (2003). Random matrices, non-colliding particle systems and queues. Séminaire de Probabilités XXXVI . Lecture Notes in Math. 1801 165–182. Springer, Berlin. · Zbl 1041.15019 [23] O’Connell, N. and Yor, M. (2001). Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96 285–304. · Zbl 1058.60078 [24] Petrov, V. V. (1995). Limit Theorems of Probability Theory . Oxford Univ. Press. · Zbl 0826.60001 [25] Rost, H. (1981). Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41–53. · Zbl 0451.60097 [26] Seppäläinen, T. (1998). Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Probab. 26 1232–1250. · Zbl 0935.60093 [27] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Etudes Sci. Publ. Math. 81 73–205. · Zbl 0864.60013 [28] van den Berg, J. and Kesten, H. (1993). Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 3 56–80. JSTOR: · Zbl 0771.60092
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.