Year: 2026
Type: preprint
Abstract: Determining the number of realisations, up to isometries, of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that, asymptotically almost surely, the $d$-dimensional complex realisation number for each $n$-vertex graph in an Erdős-Rényi random graph process is either infinite or equal to $2^{n-t}$ where $t$ is the size of the $(d+1)$-core; moreover this number coincides exactly with the real realisation number for such graphs. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ positive semi-definite matrix completion problem with randomly selected non-diagonal unknown entries.
Source: arXiv (Cornell University)
Authors: Dewar, Sean, Nixon, Anthony, Smith, Ben
Institutions: KU Leuven, Lancaster University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Exploiting combinatorial structures for algebraic and geometric decompositions, Sample complexity for Gaussian graphical models and liftability of geometric structures, Abstract rigidity for natural stability problems, G4869584417, G4872507874, G562246749, Matroids in Applied and Combinatorial Commutative Algebra, G7323488489, G7618753153
OpenAlex ID: W7161687726