The Lyons–White conjecture: rate-monotonicity of $\ell^{2m}$ distances for random walks on dihedral groups

No such pair exists: for every positive integer $m$ and every $n$, $(D_n,2m)$ is rate-monotonic (Theorem A), and more generally so is every inversion extension of a finite abelian group by an involution, a family containing the generalized dihedral, dicyclic and generalized quaternion groups (Theorem B). The picture is completed in the other direction: for every real $p\ge 1$ that is not an even integer there is an $n$ with $(D_n,p)$ not rate-monotonic (Theorem C), so the even integers are exactly the exponents for which monotonicity holds on the dihedral groups.

No such pair exists: for every positive integer $m$ and every $n$, $(D_n,2m)$ is rate-monotonic (Theorem A), and more generally so is every inversion extension of a finite abelian group by an involution, a family containing the generalized dihedral, dicyclic and generalized quaternion groups (Theorem B). The picture is completed in the other direction: for every real $p\ge 1$ that is not an even integer there is an $n$ with $(D_n,p)$ not rate-monotonic (Theorem C), so the even integers are exactly the exponents for which monotonicity holds on the dihedral groups.