Random matchings in linear hypergraphs

For a given hypergraph H and a vertex v∈ V(H), consider a random matching M chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if H is a d-regular linear k-uniform hypergraph, the probability that M does not cover v is (1 + od(1))d-1/k for all vertices v∈ V(H). This conjecture was proved for $k = 2$ by Kahn and Kim in $1998.$ In this paper, we disprove this conjecture for all k ≥ 3. For infinitely many values of $d,$ we construct d-regular linear k-uniform hypergraph H containing two vertices v₁ and v₂ such that P(v₁ ∉ M) = 1 - (1 + od(1))dᵏ⁻² and P(v₂ ∉ M) = (1 + od(1))/d+1. The gap between P(v₁ ∉ M) and P(v₂ ∉ M) in this H is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil's result on matching polynomials and paths in graphs, which is of independent interest.