Effective Mixing and Counting in Bruhat-Tits Trees

Let $\mathcal{T}$ be a locally finite tree, $\mathrm{\Gamma }$ be a discrete subgroup of $\text{Aut}(\mathcal{T})$ and $\tilde{F}$ be a $\mathrm{\Gamma }$-invariant potential. Suppose that the length spectrum of $\mathrm{\Gamma }$ is not arithmetic. In this case, we prove the exponential mixing property of the geodesic translation map $\varphi \text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\mathrm{\backslash }S\mathcal{T}\to \mathrm{\Gamma }\mathrm{\backslash }S\mathcal{T}$ with respect to the measure $m_{\mathrm{\Gamma },F}^{{\nu }^{-},{\nu }^{+}}$ under the assumption that $\mathrm{\Gamma }$ is full and $(\mathrm{\Gamma },\tilde{F})$ has weighted spectral gap property. We also obtain the effective formula for the number of $\mathrm{\Gamma }$-orbits with weights in a Bruhat-Tits tree $\mathcal{T}$ of an algebraic group.