Long-term stability of interacting Hawkes processes on random graphs

We consider a population of Hawkes processes modeling the activity of $N$ interacting neurons. The neurons are regularly positioned on the segment $[0,1]$, and the connectivity between neurons is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices through a spatial kernel. The main result of the paper concerns the longtime stability of the synaptic current of the population, as $N\to \mathrm{\infty }$, in the subcritical regime in case the synaptic memory kernel is exponential, up to time horizons that are polynomial in $N$.