Particle Systems with Local Interactions via Hitting Times and Cascades on Graphs

We study particle systems interacting via hitting times on sparsely connected graphs, following the framework of Lacker, Ramanan and Wu (2023). We provide general robustness conditions that guarantee the well-posedness of physical solutions to the dynamics, and demonstrate their connections to the dynamic percolation theory. We then study the limiting behavior of the particle systems, establishing the continuous dependence of the joint law of the physical solution on the underlying graph structure with respect to local convergence and showing the convergence of the global empirical measure, which extends the general results by Lacker et al. to systems with singular interaction. The model proposed provides a general framework for analyzing systemic risks in large sparsely connected financial networks with a focus on local interactions, featuring instantaneous default cascades.