Abstract
We study a multivariate, non-linear Hawkes process ZN on the complete graph with N nodes. Each vertex is either excitatory (probability p) or inhibitory (probability 1−p). We take the mean-field limit of ZN, leading to a multivariate point process Z̄. If p≠12, we rescale the interaction intensity by N and find that the limit intensity process solves a deterministic convolution equation and all components of Z̄ are independent. In the critical case, p=12, we rescale by N1/2 and obtain a limit intensity, which solves a stochastic convolution equation and all components of Z̄ are conditionally independent.
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