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 $\mathrm{\Gamma }\mathrm{\backslash }\mathcal{T}$ a locally finite tree. Main examples of such graphs are quotients of a Bruhat-Tits tree $\mathcal{T}$ by non-cocompact discrete subgroups $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }\mathrm{\backslash }\mathcal{T}$. Namely, for Bowen-Margulis measure $\mu$ on the space $\mathrm{\Gamma }\mathrm{\backslash }\mathcal{G}\mathcal{T}$ of geodesics and the critical exponent $\delta$ of $\mathrm{\Gamma }$, we show that there exists a constant $C$ depending on $\mathrm{\Gamma }$ and $D$ such that $$\underset{T\to \mathrm{\infty }}{\lim }\mu \left(\{[l]\in \mathrm{\Gamma }\mathrm{\backslash }\mathcal{G}\mathcal{T}\text{\hspace{0.17em}⁣}:d(D,l(t))\le N+y\}\right)=e^{-q^y\mathrm{/}{e}^{2\delta y}}$$ with $$N={\log }_{e^{2\delta \mathrm{/}q}}\left(\frac{T(e^{2\delta -q)}}{2e^{2\delta }-C(e^{2\delta }-q)}\right)\mathrm{.}$$