An Exact Obstruction to Uniform Average Mixing on $P_{11}$

Abstract

We prove that the path $P_{11}$ does not admit uniform average mixing under any probability distribution on $\mathbb{R}$, answering a question of Baptista, Coutinho, and Marques in the negative. The proof is exact: we construct an explicit rational symmetric matrix $Y$ such that $⟨Y,M(t){⟩}_F=1$ for every $t\in \mathbb{R}$, whereas $⟨Y,J/11{⟩}_F=12/11$.

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