Normal approximation for statistics of randomly weighted complexes

We prove normal approximation bounds for statistics of randomly weighted (simplicial) complexes. In particular, we consider the complete d-dimensional complex on n vertices with d-simplices equipped with i.i.d. weights. Our normal approximation bounds are quantified in terms of stabilization of difference operators, i.e., the effect on the statistic under addition/deletion of simplices. Our proof is based on Chatterjee's normal approximation bound and is a higher-dimensional analogue of the work of Cao on sparse Erdos-Renyi random graphs but our bounds are more in the spirit of 'quantitative two-scale stabilization' bounds by Lachieze-Rey, Peccati, and Yang. As applications, we prove a CLT for nearest face-weights in randomly weighted dcomplexes and give a normal approximation bound for local statistics of random d-complexes.