Optimal local-expansion constant for path-complex trickle-down

Determine whether the optimal local spectral-expansion constant in the trickle-down theorem for connected d-partite path complexes that are 1/2-top-link spectral expanders is α=1/2.

Background

The paper proves that every connected d-partite path complex whose codimension-two links have second random-walk eigenvalue at most 1/2 is an α-local spectral expander for some α≤π/2−1. The authors establish a lower-bound construction showing that the top-link threshold 1/2 is tight, but their upper bound on the resulting local-expansion parameter is weaker than 1/2.

The unresolved issue is whether the local-expansion constant can be improved from π/2−1 to the conjecturally optimal value 1/2. Resolving this would sharpen the principal trickle-down theorem and improve the associated sampling and mixing-time bounds.

References

We conjecture that the optimal α in the above theorem is α=1/2.

— Optimal Trickle-Down Theorems for Path Complexes via C-Lorentzian Polynomials with Applications to Sampling and Log-Concave Sequences  (2503.01005 - Leake et al., 2 Mar 2025) in Section 1, immediately after Theorem 1 (Trickle-down Theorem for Path Complexes)