On the joint behaviour of speed and entropy of random walks on groups

For every $3/4\le δ, β< 1$ satisfying $δ\leq β< \frac{1+δ}{2}$ we construct a finitely generated group $Γ$ and a (symmetric, finitely supported) random walk $X_n$ on $Γ$ so that its expected distance from its starting point satisfies $E|X_n|\asymp n^β$ and its entropy satisfies $H(X_n)\asymp n^δ$. 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 $n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^γ$ for some $γ<1$.

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