Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model. (Q1879491)

The Richardson model is a~particular case of interacting particle systems. The two-type Richardson model is a~system with the three states \(\{0,1,2\}\) and the evolution described by the following flip rates: 1's and 2's never flip, while 0 flips to 1 (resp. 2) at rate \(\lambda _1\) \((\lambda _2)\) times the number of nearest neighbors with value 1 (2). The state 1 (2) at site \(x\in {\mathbb Z}^{d}\) means occupation by a~particle of the first (second) type, and once the site being occupied, it cannot be changed. Starting with only one particle of each type, there are only three possible scenarios: (i) The set of 1's surrounds the (finite) set of 2's. (ii) The set of 2's surrounds the (finite) set of 1's. (iii) Both the sets of 1's and 2's grow indefinitely. The problem to be solved is whether the scenario (iii) can happen with positive probability. In a~previous paper the authors proved that for \(d=2\) and \(\lambda _1=\lambda _2\), (iii) really occurs with positive probability. In the present paper they conjecture that (iii) has zero probability whenever \(\lambda _1\neq \lambda _2\). As the main result of the paper they manage to prove the conjecture for \(\lambda _1=1\) and all but countably many values of \(\lambda _2\).