Coexistence for Richardson type competing spatial growth models
By Christopher E. Hoffman. 2004
104 citations recorded by the source corpus.
We study a large family of competing spatial growth models. In these the vertices in Z^d can take on three possible states {0,1,2}. Vertices in states 1 and 2 remain in their states forever, while vertices in state 0 which are adjacent to a vertex in state 1 (or state 2) can switch to state 1 (or state 2). We think of the vertices in states 1 and 2 as infected with one of two infections while the vertices in state 0 are considered uninfected. In this way these models are variants of the Richardson model. We start the models with a single vertex in state 1 and a single vertex is in state 2. We show that with positive probability state 1 reaches an infinite number of vertices and state 2 also reaches an infinite number of vertices. This extends results and proves a conjecture of Haggstrom and Pemantle. The key tool is applying the ergodic theorem to stationary first passage percolation.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/5a743d21961851a77ec29d8304ce2678ed7092d4
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In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Stationary cocycles and Busemann functions for the corner growth model (2015, methodology): “Distinct from this line of work is Hoffman’s use of Busemann functions to study competition under more gener al weight distributions [31, 32].”
- Geodesics in first passage percolation (2005, unknown): “Garet and Marchand [7] and Hoffman [12] proved that in any dimension mutual unbounded growth is possible when k = 2.”
- 50 years of first passage percolation (2015, background): “Shortly thereafter, Garet and Marchand [79] and Hoffman [101] independently extended these results to prove coexistence of 2 species for a broad class of translation-invariant measures, including some non-i.i.d. ones.”