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

We study a multivariate, non-linear Hawkes process Z N on a q-Erdős-Rényi-graph with N nodes. Each vertex is either excitatory (probability p) or inhibitory (probability 1 − p). If p = 1 2 , we take the mean-field limit of Z N , leading to a multivariate point process Z. We rescale the interaction intensity by N and find that the limit intensity process solves a deterministic convolution equation and all components of Z are independent. The fluctuations around the mean field limit converge to the solution of a stochastic convolution equation. In the critical case, p = 1 2 , we rescale by N 1/2 and discuss difficulties, both heuristically and numerically.

We study once-reinforced random walk on Z. For this model, we derive limit results on all moments of its range using Tauberian theory.

We study once-reinforced random walk (ORRW) on Z. For this model, we derive limit results on all moments of its range using Tauberian theory.