Limiting distribution of geodesics in a geometrically finite quotients of regular trees

In this article, we prove an extreme value theorem on the limit distribution of geodesics in a geometrically finite quotient of $\Gamma\backslash\mathcal{T}$ a locally finite tree. Main examples of such graphs are quotients of a Bruhat-Tits tree $\mathcal{T}$ by non-cocompact discrete subgroups $\Gamma$ of $PGL(2,\mathbf{K})$ of a positive characteristic local field $\mathbf{K}$. We investigate, for a given time $T$, the measure of the set of $\Gamma$-equivalent geodesic classes which stay up to time $T$ the region of distance $d$ at most $N$ depending on $T$ from a fixed compact subset $D$ of $\Gamma\backslash\mathcal{T}$. Namely, for Bowen-Margulis measure $\mu$ on the space $\Gamma\backslash\mathcal{GT}$ of geodesics and the critical exponent $\delta$ of $\Gamma$, we show that there exists a constant $C$ depending on $\Gamma$ and $D$ such that $$\lim_{T\to\infty}\mu\left(\left\{[l]\in\Gamma\backslash\mathcal{GT}\colon \underset{0\le t \le T}{\textrm{max}}d(D,l(t))\le N+y\right\}\right)=e^{-q^y/e^{2\delta y}}$$ with $$N=\log_{e^{2\delta/q}}\left(\frac{T(e^{2\delta-q)}}{2e^{2\delta}-C(e^{2\delta}-q)}\right).$$