Random chain complexes of real vector spaces

We introduce a natural class of models of random chain complexes of real vector spaces that some classical ensembles of random matrices, the length $1$ case. We are interested here in the homological properties of these random complexes. For chain complexes of length $1$ or $2$, we characterize the Betti numbers almost surely, in terms of the dimensions of the vector spaces. We further examine complexes of length $3$ with some constraints on dimensions, as well as complexes of arbitrary finite length in which all vector spaces have equal dimension. Across all these settings, we show that the sum of the Betti numbers is almost surely as small as possible, attaining a trivial lower bound $|χ|$ dictated by the dimensions of the underlying vector spaces and the Euler formula. These results suggest an underlying algebraic heuristic for a phenomenon frequently observed in stochastic topology, that nontrivial homology rarely appears unless forced to.

💡 Research Summary

The paper introduces a probabilistic framework for random chain complexes built from real vector spaces. For a fixed sequence of dimensions ( \mathbf a = (a_0,\dots,a_n) ), each boundary map ( d_i : A_{i-1}\to A_i ) is taken as a random real matrix drawn from a continuous density (e.g., a multivariate Gaussian). Because Lebesgue measure cannot be used to condition on events of probability zero, the authors define conditional probabilities via Hausdorff measures and ε‑thickening of algebraic varieties, allowing rigorous treatment of “conditioning on a zero‑probability event.”

The existence of a complex with prescribed ranks ( \mathbf r = (r_1,\dots,r_n) ) is characterized by the simple linear constraints ( r_i + r_{i+1} \le a_i ) for all ( i ). When these constraints hold, the variety of complexes with those ranks has dimension
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