Long-range competition on the torus

We study competition between two growth models with long-range correlations on the torus $\mathbb T_n^d$ of size $n$ in dimension $d$. We append the edge set of the torus $\mathbb T_n^d$ by including all non-nearest-neighbour edges, and from two source vertices $v^\ominus$ and $v^\oplus$ in $\mathbb T_n^d$ two infection processes $\ominus$ and $\oplus$ start spreading to other vertices. Each susceptible vertex can be infected by at most one infection type and when infected stays infected forever (i.e.\ competing SI models). A vertex $v$ infected with type $\square\in\{\ominus,\oplus\}$ infects a susceptible vertex $u$ at rate $\lambda_\square \|u-v\|^{-\alpha_\square}$, where $\lambda_{\ominus}=\lambda_\ominus(n),\lambda_\oplus=\lambda_\oplus(n)>0$ and $\alpha_\ominus=\alpha_\ominus(n),\alpha_\oplus=\alpha_\oplus(n)\in[0,d)$ are allowed to depend on $n$. We study coexistence, the event that both infections reach an asymptotically positive proportion of the graph as $n$ tends to infinity, and identify precisely when coexistence occurs. In the case of absence of coexistence, we outline several phase transitions in the size of the infection that reaches a negligible proportion of the vertices, which depends on the ratio of the sum of infection rates across all vertices of type $\ominus$ and $\oplus$. The work extends known results for the case $\alpha_\ominus(n)=\alpha_\oplus(n)\equiv 0$ and $\lambda_\ominus(n)\equiv 1, \lambda_\oplus(n)\equiv \lambda>0$, and includes general and novel results that cannot be observed when the model parameters are fixed and independent of $n$. The main technical contribution is a coupling of the competition process with branching random walks, where we are able to use the coupling even when coupling error between the competition process and the branching random walks is of the same order of magnitude as the size of the coupled processes.