When Do Two Distributions Yield the Same Expected Euler Characteristic Curve in the Thermodynamic Limit?

Let $F$ be a probability distribution on $\mathbb{R}^d$ which admits a bounded density. We investigate the Euler characteristic of the \v{C}ech complex on $n$ points sampled from $F$ i.i.d. as $n\to\infty$ in the thermodynamic limit regime. As a main result, we identify a condition for two probability distributions to yield the same expected Euler characteristic under this construction. Namely, this happens if and only if their densities admit the same excess mass transform. Building on work of Bobrowski, we establish a connection between the limiting expected Euler characteristic of any such probability distribution $F$ and the one of the uniform distribution on $[0,1]^d$ through an integral transform. Our approach relies on constructive proofs, offering explicit calculations of expected Euler characteristics in lower dimensions as well as reconstruction of a distribution from its limiting Euler characteristic. In the context of topological data analysis, where the Euler characteristic serves as a summary of the shape of data, we address the inverse problem and determine what can be discriminated using this invariant. This research sheds light on the relationship between a probability distribution and topological properties of the \v{C}ech complex on its samples in the thermodynamic limit.