Moderate deviations and laws of the iterated logarithm for geometric functionals in the sparse regime

We prove moderate deviation principles and laws of the iterated logarithm in the sparse regime for geometric and topological functionals associated with $k$-point connected components. These limit theorems are established at both the measure-valued and vector-valued levels, for Poisson and binomial point processes. As applications, we derive corresponding results for component counts in random geometric graphs and Čech complexes and for counts of Morse critical points.