Project DetailsIn this project, we describe the phase transition of the homological connectivity of a random complex, constructed on a bounded manifold, which was chosen to be a flat cylinder of a general dimension. We do this by generating a random cloud on the cylinder, on which we construct a filtration of a simplicial complex, chosen to be the ech Complex. We try to describe the phase transition of the homological connectivity of the complex i.e., the transition in which the homology of the random complex becomes identical to that of the underlying cylinder. We expect the addition of the boundary (which isn't trivial) to introduce a significant change to the threshold. We find this phase transition by investigating the critical faces of the complex Those are simplices that delineate change in the homology of the complex when they are created or destroyed. Our work focuses on finding the average number (expectation) of those critical faces.
To solve this problem, we divide the cylinder into two sections An inner section that does not interact with the boundary, and a boundary section that includes two stripes, each includes one of the two boundary lines of the cylinder. With the help of integration on the spheres of the complex, we find the average number of critical faces in each section. Whereas the phase transition in the inner section happens exactly when it would happen on a manifold without boundary, we find the threshold in the boundary section is roughly twice as large as the threshold in the inner area This is the significant change that we expected the boundary to introduce. To achieve a more precise threshold, we choose to focus on two private cases in a general dimension d When the critical faces are 1-dimensional, and when they are d-dimensional. Finally, we use a Matlab simulation to validate our assumptions and results.
