The winner takes it all

We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate $\lambda_1$ ($\lambda_2$) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent $\tau\in(2,3)$, then, as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of $\lambda_1$ and $\lambda_2$. The picture is similar with multiple starting points for the infections.

[1]  D. R. Grey.  Explosiveness of age-dependent branching processes , 1974 .

[2]  Robin Pemantle,et al.  First passage percolation and a model for competing spatial growth , 1997 .

[3]  Russell Lyons,et al.  Random walk in a random environment and rst-passage percolation on trees , 1992 .

[4]  Svante Janson,et al.  A new approach to the giant component problem , 2007, Random Struct. Algorithms.

[5]  C. Newman,et al.  Divergence of Shape Fluctuations in Two Dimensions , 1995 .

[6]  Olivier Garet,et al.  Coexistence in two-type first-passage percolation models , 2003 .

[7]  D. Richardson.  Random growth in a tessellation , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  J. T. Cox,et al.  Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions , 1981 .

[9]  G. Hooghiemstra,et al.  First passage percolation on random graphs with finite mean degrees , 2009, 0903.5136.

[10]  H. Kesten.  On the Speed of Convergence in First-Passage Percolation , 1993 .

[11]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[12]  Russell Lyons,et al.  Correction: Random walk in a random environment and first-passage percolation on trees , 1992 .

[13]  R. Smythe,et al.  First-passage percolation on the square lattice. I , 1977, Advances in Applied Probability.

[14]  Bruce A. Reed,et al.  The Size of the Giant Component of a Random Graph with a Given Degree Sequence , 1998, Combinatorics, Probability and Computing.

[15]  Christopher Hoffman.  Coexistence for Richardson type competing spatial growth models , 2004 .

[16]  Olle Häggström,et al.  Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model , 2000 .

[17]  Remco van der Hofstad,et al.  First Passage Percolation on the Erdős–Rényi Random Graph , 2010, Combinatorics, Probability and Computing.

[18]  Svante Janson.  The Probability That a Random Multigraph is Simple , 2009, Comb. Probab. Comput..

[19]  Remco van der Hofstad,et al.  Universality for first passage percolation on sparse random graphs , 2012, 1210.6839.

[20]  Remco van der Hofstad,et al.  Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds , 2014, 1408.0475.

[21]  Harry Kesten,et al.  First-passage percolation, network flows and electrical resistances , 1984 .

[22]  J. Hammersley,et al.  First-Passage Percolation, Subadditive Processes, Stochastic Networks, and Generalized Renewal Theory , 1965 .

[23]  Remco van der Hofstad,et al.  A Phase Transition for the Diameter of the Configuration Model , 2007, Internet Math..