  For every $3/4\le \delta, \beta< 1$ satisfying $\delta\leq \beta <
\frac{1+\delta}{2}$ we construct a finitely generated group $\Gamma$ and a
(symmetric, finitely supported) random walk $X_n$ on $\Gamma$ so that its
expected distance from its starting point satisfies $E|X_n|\asymp n^{\beta}$
and its entropy satisfies $H(X_n)\asymp n^\delta$. In fact, the speed and
entropy can be set precisely to equal any two nice enough prescribed functions
$f,h$ up to a constant factor as long as the functions satisfy the relation
$n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq
n^\gamma$ for some $\gamma<1$.
