Some characteristics of the second betti number of random two dimensional simplicial complexes
Kang Tan · cIRcle (University of British Columbia) · 2009
In this thesis, through generating random two-dimensional simplicial complexes, we studied the event (b₂=0) for some specific probabilities. We found when the probability of event (b₂=0) takes on certain specific values, the pair (n₀,n₂) lies on certain lines. However, this research is limited by our sample space ( i.e. for P(b₂=0) ≈ 10%, 10 ≤ n₀ ≤ 80; for P(b₂=0) ≈ 50%, 12 ≤ n₀ ≤ 100; for P(b₂=0) ≈ 90%, 12 ≤ n₀ ≤ 145). The " linear behavior" may not hold asymptotically. In the same time, we endeavor to find the number of tetrahedra and 6-triangles in the simplicial complexes. When the event (b₂=0) occurs in our specific probabilities, it seems the second Betti number should come from tetrahedra and 6-triangles with high probability. However, the expectation of the number of tetrahedra and 6-triangles goes to zero, when no goes to infinity and there exists linear relationships between the pair (n₀,n₂). This evidence may also support that the " linear behavior" may not hold asymptotically. If n₂ and n₀ vary linearly with n₀ going to infinity, then the probability that n₀(Κ) — n₀ is extremely small in model MB for reasons are similar to the Coupon Collector's problem. Hence, the probability that we cannot find element in S(n₀,n₂) is large, which indicates that the model may have problems.