The wreath product of $\mathbb{Z}$ with $\mathbb{Z}$ has Hilbert compression exponent $\frac{2}{3}$

Let $G$ be a finitely generated group, equipped with the word metric $d$ associated with some finite set of generators. The Hilbert compression exponent of $G$ is the supremum over all $\alpha\ge 0$ such that there exists a Lipschitz mapping $f:G\to L_2$ and a constant $c>0$ such that for all $x,y\in G$ we have $\Vert f(x)-f(y)\Vert _2\ge cd(x,y)^\alpha.$ It was previously known that the Hilbert compression exponent of the wreath product $\mathbb{Z}\,{\boldsymbol{\wr}}\, \mathbb{Z}$ is between $\frac23$ and $\frac34$. Here we show that $\frac23$ is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.