2024/03/08 by Bas Lodewijks, Neeladri Maitra, Lodewijks, Bas +1
Economics, Econometrics and Finance · Engineering · #Adhesion, Friction, and Surface Interactions #FOS: Mathematics #Merger and Competition Analysis #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2403.05536
openalex publication_date 2024/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study competition between two growth models with long-range correlations on the torus \mathbb Tnd of size n in dimension d. We append the edge set of the torus \mathbb Tnd by including all non-nearest-neighbour edges, and from two source vertices v^\ominus and v^⊕ in \mathbb Tnd two infection processes \ominus and ⊕ 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∈\\ominus,⊕\ infects a susceptible vertex u at rate λ_\square ‖u-v‖-α_\square, where λ\ominus=λ_\ominus(n),λ_⊕=λ_⊕(n)>0 and α_\ominus=α_\ominus(n),α_⊕=α_⊕(n)∈[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 ⊕. The work extends known results for the case α_\ominus(n)=α_⊕(n)≡ 0 and λ_\ominus(n)≡ 1, λ_⊕(n)≡ λ>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.