Continuum tree limit for the range of random walks on regular trees

Let $b$ be an integer greater than 1 and let $W^{\text{ee}}=(W_n^{\text{ee}};n\ge 0)$ be a random walk on the $b$-ary rooted tree ${\text{U}}_b$, starting at the root, going up (resp. down) with probability $1\mathrm{/}2+ϵ$ (resp. $1\mathrm{/}2-ϵ$), $ϵ\in (0,1\mathrm{/}2)$, and choosing direction $i\in \{1,\mathrm{.}\mathrm{.}\mathrm{.},b\}$ when going up with probability $a_i$. Here $\mathring{a}=(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_b)$ stands for some non-degenerated fixed set of weights. We consider the range ${W^{\ee}n ; n\geq 0 }$thatisasubtreeof$\U_b $. It corresponds to a unique random rooted ordered tree that we denote by ${\tau }_{ϵ}$. We rescale the edges of ${\tau }_{ϵ}$ by a factor $\text{ee}$ and we let $\text{ee}$ go to 0: we prove that correlations due to frequent backtracking of the random walk only give rise to a deterministic phenomenon taken into account by a positive factor $\gamma (\aa)$\mathrm{.}Moreprecisely,weprovethat$\tau{\epsilon}$ converges to a continuum random tree encoded by two independent Brownian motions with drift conditioned to stay positive and scaled in time by $\gamma (\mathring{a})$. We actually state the result in the more general case of a random walk on a tree with an infinite number of branches at each node ($b=\mathrm{\infty }$) and for a general set of weights $\mathring{a}=(a_n,n\ge 0)$.