Limiting distribution for intersections of conjugate Sylow 2-subgroups in Sn

Determine the limiting distribution, as n tends to infinity, of the random variable |Pn ∩ Pn^{x}| where Pn is a Sylow 2-subgroup of the symmetric group Sn and x is chosen uniformly at random from Sn; in particular, ascertain whether the limiting probability f(n,2)=P(|Pn ∩ Pn^{x}|>1) converges to 1−e^{-1/2}.

Background

The paper studies double cosets Pn \ Sn / Pn where Pn is a Sylow p-subgroup of Sn, with a focus on the sizes and distribution of these double cosets. For p=2, Theorem 1.1(b) shows that the probability f(n,2) that two conjugate Sylow 2-subgroups intersect nontrivially is bounded away from zero.

Section 4.2 analyzes an auxiliary model involving elementary abelian 2-subgroups associated with fixed-point-free involutions, proving that the number of matching pairs between two random perfect matchings has a Poisson(1/2) limit, yielding a lower bound ~1−e^{-1/2} for P(|Pn ∩ Pn^{x}|>1). However, the authors do not identify the limiting distribution of the actual intersection size |Pn ∩ Pn^{x}|.

In Section 6, the authors note they believe the lower bound may be sharp for f(n,2) but explicitly state that the limiting distribution of |Pn ∩ Pn^{x}| remains unknown.