The dimension of harmonic measure on Galton–Watson trees increases at small bias

Let $T$ be a Galton–Watson tree without leaves whose offspring number $N$ is not constant and has finite mean $m$, and let $d_\lambda$ be the Hausdorff dimension of the harmonic measure of the $\lambda$-biased random walk on $T$, $0\le\lambda<m$. Lyons, Pemantle and Peres asked whether $d_\lambda$ increases with $\lambda$. We prove that $\lambda\mapsto d_\lambda$ is continuously differentiable on $[0,k_0)$, where $k_0$ is the smallest value taken by $N$, and that \[ \frac{d}{d\lambda}d_\lambda\Big|_{\lambda=0}=-\bigl(1-\mathbb{E}[1/N]\bigr)\operatorname{Cov}\bigl(\log N,1/N\bigr)>0 . \] Hence $d_\lambda$ is strictly increasing on an interval $[0,\varepsilon]$, with $\varepsilon>0$ depending on the offspring law. No moment assumption beyond the finite mean is needed. The proof combines the formula of Lin and Rousselin for $d_\lambda$ with a contraction estimate, uniform over trees, for the recursion satisfied by the escape probabilities and by their derivatives in $\lambda$.