Proof. (i) By (72) and (74), we have that

$$
\begin{align*}
|\mathbb{E}[\log \rho_x]| &= \left| \psi\left(\frac{(|x| + na_n^{-1})^\alpha}{2\Delta_n}\right) - \psi\left(\frac{((|x|-1) + na_n^{-1})^\alpha + \Delta_n}{2\Delta_n}\right) \right| \\
&\le \log\left(\left|\frac{(|x| + na_n^{-1})^\alpha}{((|x|-1) + na_n^{-1})^\alpha + \Delta_n}\right|\right) + \frac{2\Delta_n}{((|x|-1) + na_n^{-1})^\alpha + \Delta_n} \\
&\le \frac{2(|x| + na_n^{-1})^\alpha - ((|x|-1) + na_n^{-1})^\alpha + 4\Delta_n}{(|x| + na_n^{-1})^\alpha + \Delta_n} \le \frac{2\alpha}{(|x| + na_n^{-1})} + \frac{4\Delta_n}{(|x| + na_n^{-1})^\alpha}.
\end{align*}
$$

(ii) First note that it follows from (72) that for all $t \in [0, 1]$,

$$
\begin{align}
\sum_{O_n < x \le x_{[2nt]}} \mathbb{E}[\log(\rho_x)] &= \sum_{O_n < x \le x_{[2nt]}} \left\{ \psi\left(\frac{((|x|-1) + na_n^{-1})^\alpha + \Delta_n}{2\Delta_n}\right) - \psi\left(\frac{(|x| + na_n^{-1})^\alpha}{2\Delta_n}\right) \right\} \nonumber \\
&= \psi\left(\frac{(na_n^{-1})^\alpha + \Delta_n}{2\Delta_n}\right) - \psi\left(\frac{(|x_{[2nt]}| + na_n^{-1})^\alpha}{2\Delta_n}\right) \nonumber \\
&\quad + \sum_{i=1}^{|x_{[2nt]}|-1} \left\{ \psi\left(\frac{(i + na_n^{-1})^\alpha}{2\Delta_n} + \frac{1}{2}\right) - \psi\left(\frac{(i + na_n^{-1})^\alpha}{2\Delta_n}\right) \right\}. \tag{76}
\end{align}
$$

First, we assume that $\alpha \le 1$. For the first term in (76), we use (74) and note that for almost every
$\omega \in \Omega$,

$$
\begin{align*}
\log \left( \frac{(na_n^{-1})^\alpha + \Delta_n}{2\Delta_n} \right) - \log \left( \frac{(|x_{[2nt]}| + na_n^{-1})^\alpha}{2\Delta_n} \right) &= \log \left( \frac{(na_n^{-1})^\alpha + \Delta_n}{(|x_{[2nt]}| + na_n^{-1})^\alpha} \right) \\
&= -\alpha \log(d(O, t) + 1) + o(1),
\end{align*}
$$

as $n \to \infty$ by (9). Thus, by (74), for almost every $\omega \in \Omega$ the first term in (76) satisfies

$$
\left| \psi \left( \frac{(na_n^{-1})^\alpha + \Delta_n}{2\Delta_n} \right) - \psi \left( \frac{(|x_{[2nt]}| + na_n^{-1})^\alpha}{2\Delta_n} \right) + \alpha \log(d(O, t) + 1) \right| \le \frac{2\Delta_n}{(na_n^{-1})^\alpha} + o(1),
$$

as $n \to \infty$, and therefore

$$
\psi \left( \frac{(na_n^{-1})^\alpha + \Delta_n}{2\Delta_n} \right) - \psi \left( \frac{(|x_{[2nt]}| + na_n^{-1})^\alpha}{2\Delta_n} \right) = -\alpha \log(d(O, t) + 1) + o(1),
$$

as $n \to \infty$.

For the second term, since $\inf_{i \ge 0} \frac{(n^{-1}a_n+i)^\alpha}{2\Delta_n} \to \infty$ as $n \to \infty$, it follows from (73) that for all $i \ge 0$,

$$
\psi \left( \frac{(i + na_n^{-1})^{\alpha}}{2\Delta_n} + \frac{1}{2} \right) - \psi \left( \frac{(i + na_n^{-1})^{\alpha}}{2\Delta_n} \right) = \frac{\Delta_n}{(i + na_n^{-1})^{\alpha}} (1 + o(1)),
$$

where the $o(1)$ term is uniform over all $i \ge 0$ (but may depend on $\omega$).

This implies that for almost every $\omega \in \Omega$ it holds for all $t \in [0, 1]$ that

$$
\sum_{i=na_n^{-1}+1}^{|x_{[2nt]}|-1+na_n^{-1}} \left\{ \psi\left(\frac{i^\alpha}{2\Delta_n} + \frac{1}{2}\right) - \psi\left(\frac{i^\alpha}{2\Delta_n}\right) \right\} \\
\sim \sum_{i=na_n^{-1}+1}^{|x_{[2nt]}|-1+na_n^{-1}} \frac{\Delta_n}{i^\alpha}
$$
$$ \sim \begin{cases} \frac{\Delta_n}{1-\alpha} \left[ (|x_{\lfloor 2nt \rfloor}| - 1 + na_n^{-1})^{1-\alpha} - (na_n^{-1})^{1-\alpha} \right] & \text{if } \alpha < 1, \\ \Delta \left[ \log(|x_{\lfloor 2nt \rfloor}| - 1 + na_n^{-1}) - \log(na_n^{-1}) \right] & \text{if } \alpha = 1, \end{cases} $$

$$ \sim \begin{cases} \frac{\Delta}{1-\alpha} \left[ \left[ \frac{(|x_{\lfloor 2nt \rfloor}| - 1) + na_n^{-1}}{na_n^{-1}} \right]^{1-\alpha} - 1 \right] & \text{if } \alpha < 1, \\ \Delta \log(n^{-1}a_n|x_{\lfloor 2nt \rfloor}| - n^{-1}a_n + 1) & \text{if } \alpha = 1, \end{cases} $$

$$ \sim \begin{cases} \frac{\Delta [(d(0,t) + 1)^{1-\alpha} - 1]}{1-\alpha} & \text{if } \alpha < 1, \\ \Delta \log(d(0,t) + 1) & \text{if } \alpha = 1. \end{cases} $$

Since $\sup_{t \in [0,1]} d(0,t)$ is bounded for almost every $\omega \in \Omega$, this implies the result. $\square$

<figure><img src="image_4.png" /></figure>

### A.3 Claim 4.4: variance of the potential when $\alpha \le 1$

**Proposition A.2.** For almost every $\omega \in \Omega$, it holds uniformly over $t \in [0, 1]$ as $n \to \infty$ that:

$$ \sum_{O_n < x \le x_{\lfloor 2nt \rfloor}} \mathrm{Var}[\log(\rho_x)] \to \frac{4\Delta[(d(O,t) + 1)^{1-\alpha} - 1]}{1-\alpha} \quad \text{if } \alpha < 1, $$

$$ \sum_{O_n < x \le x_{\lfloor 2nt \rfloor}} \mathrm{Var}[\log(\rho_x)] \to 4\Delta \log(d(O,t) + 1), \quad \text{if } \alpha = 1. $$

*Proof.* It follows from (72) that for all $t \in [0, 1]$,

$$ \sum_{O_n < x \le x_{\lfloor 2nt \rfloor}} \mathrm{Var}(\log \rho_x) = \sum_{i=1}^{|x_{\lfloor 2nt \rfloor}|} \left\{ \psi' \left( \frac{((i-1) + na_n^{-1})^\alpha}{2\Delta_n} + \frac{1}{2} \right) + \psi' \left( \frac{(i + na_n^{-1})^\alpha}{2\Delta_n} \right) \right\}. $$

First we assume that $\alpha \le 1$. Since both arguments in the derivatives of the digamma function blow up uniformly over $i \ge 0$ as $n \to \infty$, it follows from (75) that for all $i \ge 1$,

$$ \psi' \left( \frac{(i + na_n^{-1})^\alpha}{2\Delta_n} \right) + \psi' \left( \frac{((i-1) + na_n^{-1})^\alpha}{2\Delta_n} + \frac{1}{2} \right) = \frac{4\Delta_n}{(i + na_n^{-1})^\alpha} (1 + o(1)), $$

as $n \to \infty$, where the $o(1)$ term is uniform over all $i \ge 1$. This implies that

$$ \sum_{O_n < x \le x_{\lfloor 2nt \rfloor}} \mathrm{Var}(\log \rho_x) \sim \sum_{i=1+na_n^{-1}}^{|x_{\lfloor 2nt \rfloor}|+na_n^{-1}} \frac{4\Delta_n}{i^\alpha} $$

$$ \sim \begin{cases} \frac{4\Delta_n}{1-\alpha} \left[ (|x_{\lfloor 2nt \rfloor}| + na_n^{-1})^{1-\alpha} - (1 + na_n^{-1})^{1-\alpha} \right] & \text{if } \alpha < 1, \\ 4\Delta \left[ \log(|x_{\lfloor 2nt \rfloor}| + na_n^{-1}) - \log(1 + na_n^{-1}) \right] & \text{if } \alpha = 1, \end{cases} $$

$$ \sim \begin{cases} \frac{4\Delta}{1-\alpha} \left[ \left[ \frac{|x_{\lfloor 2nt \rfloor}| + na_n^{-1}}{na_n^{-1}} \right]^{1-\alpha} - \left[ \frac{1 + na_n^{-1}}{na_n^{-1}} \right]^{1-\alpha} \right] & \text{if } \alpha < 1, \\ 4\Delta \left[ \log(n^{-1}a_n|x_{\lfloor 2nt \rfloor}| + 1) - \log(n^{-1}a_n + 1) \right] & \text{if } \alpha = 1, \end{cases} $$
$$ \sim \begin{cases} \frac{4\Delta[(d(0,t) + 1)^{1-\alpha} - 1]}{1-\alpha} & \text{if } \alpha < 1, \\ \frac{4\Delta \log(d(0,t) + 1)}{1-\alpha} & \text{if } \alpha = 1. \end{cases} $$

Again, since $\sup_{t \in [0,1]} d(0,t)$ is bounded for almost every $\omega \in \Omega$, this implies the result. $\square$

### A.4 Claim 4.5: exponential moments when $\alpha \le 1$

**Lemma A.3.** For almost every $\omega \in \Omega$, it holds for all $x \in T_n$ and all $1 \le k \le (na_n^{-1})^{1/2}$ that

$$ \mathbb{E}[e^{k \log \rho_x}] \le \exp \left\{ (1 + 3\Delta) \left( \frac{2k\alpha}{|x| + na_n^{-1}} + \frac{3k^2\Delta_n}{(|x| + na_n^{-1})^\alpha} \right) \right\}, $$

$$ \mathbb{E}[e^{k \log(\rho_x^{-1})}] \le \exp \left\{ (1 + 3\Delta) \left( \frac{2k\alpha}{|x| + na_n^{-1}} + \frac{3k^2\Delta_n}{(|x| + na_n^{-1})^\alpha} \right) \right\}. $$

*Proof.* First recall that, since

$$ \rho_x \sim \beta' \left( \frac{(|x| - 1 + na_n^{-1})^\alpha + \Delta_n}{2\Delta_n}, \frac{(|x| + na_n^{-1})^\alpha}{2\Delta_n} \right), $$

we also have that

$$ \frac{1}{\rho_x} \sim \beta' \left( \frac{(|x| + na_n^{-1})^\alpha}{2\Delta_n}, \frac{(|x| - 1 + na_n^{-1})^\alpha + \Delta_n}{2\Delta_n} \right). $$

Consequently (using the formula for the $k^{th}$ moment of the beta prime distribution), it holds for any $k \ge 1$ that

$$ \mathbb{E}[\rho_x^k] = \prod_{j=0}^{k-1} \frac{(|x| - 1 + na_n^{-1})^\alpha + (2j + 1)\Delta_n}{(|x| + na_n^{-1})^\alpha - (2j + 2)\Delta_n}, \quad (77) $$

$$ \mathbb{E}[\rho_x^{-k}] = \prod_{j=0}^{k-1} \frac{(|x| + na_n^{-1})^\alpha + 2j\Delta_n}{(|x| - 1 + na_n^{-1})^\alpha - (2j + 1)\Delta_n}. \quad (78) $$

Now note from (77) that

$$ \begin{aligned} \mathbb{E}[e^{k \log \rho_x}] &= \mathbb{E}[\rho_x^k] \\ &= \prod_{j=0}^{k-1} \frac{(|x| - 1 + na_n^{-1})^\alpha + (2j + 1)\Delta_n}{(|x| + na_n^{-1})^\alpha - (2j + 2)\Delta_n} \\ &\le \prod_{j=0}^{k-1} \left( 1 + \left[ \frac{(|x| - 1 + na_n^{-1})^\alpha - (|x| + na_n^{-1})^\alpha}{(|x| + na_n^{-1})^\alpha} + \frac{(4j + 3)\Delta_n}{(|x| + na_n^{-1})^\alpha} \right] \left[ 1 + \frac{2(2j + 2)\Delta_n}{(|x| + na_n^{-1})^\alpha} \right] \right) \\ &\le \exp \left\{ (1 + 3\Delta) \left( \frac{2k\alpha}{|x| + na_n^{-1}} + \frac{3k^2\Delta_n}{(|x| + na_n^{-1})^\alpha} \right) \right\}. \end{aligned} $$

Similarly, note from (78) that

$$ \begin{aligned} \mathbb{E}[e^{k \log(\rho_x^{-1})}] &= \mathbb{E}[(\rho_x)^{-k}] \\ &= \prod_{j=0}^{k-1} \frac{(|x| + na_n^{-1})^\alpha + 2j\Delta_n}{(|x| - 1 + na_n^{-1})^\alpha - (2j + 1)\Delta_n} \\ &\le \prod_{j=0}^{k-1} \left( 1 + \left[ \frac{(|x| + na_n^{-1})^\alpha - (|x| - 1 + na_n^{-1})^\alpha}{(|x| - 1 + na_n^{-1})^\alpha} + \frac{(4j + 1)\Delta_n}{(|x| - 1 + na_n^{-1})^\alpha} \right] \left[ 1 + \frac{2(2j + 1)\Delta_n}{(|x| + na_n^{-1})^\alpha} \right] \right) \\ &\le \exp \left\{ (1 + 3\Delta) \left( \frac{2k\alpha}{|x| + na_n^{-1}} + \frac{3k^2\Delta_n}{(|x| + na_n^{-1})^\alpha} \right) \right\}. \end{aligned} $$

$\square$
### A.5 Expectation of the potential when $\alpha > 1$

In the transient regime $\alpha > 1$ considered in Section 8, we do not rescale the initial weights so for each $x \in T_\infty$ we have

$$ \rho_x \sim \beta' \left( \frac{((|x|-1))^{\alpha} + \Delta}{2\Delta}, \frac{|x|^{\alpha}}{2\Delta} \right). $$

**Lemma A.4.** Assume $\alpha > 1$. Then for almost every $\omega \in \Omega$, the following holds. As $|y| \to \infty$, we have that

$$ \sum_{O_{\infty} \prec x \prec y} \mathbb{E}[\log \rho_x] = -\alpha \log |y| + O(1). $$

*Proof.* First note that it follows from (72) that for any $y \in T_\infty$,

$$ \begin{aligned} \sum_{O_{\infty} \prec x \prec y} \mathbb{E}[\log(\rho_x)] &= \sum_{O_{\infty} \prec x \prec y} \left\{ \psi\left(\frac{(|x|-1)^{\alpha} + \Delta}{2\Delta}\right) - \psi\left(\frac{|x|^{\alpha}}{2\Delta}\right) \right\} \\ &= \psi\left(\frac{1}{2}\right) - \psi\left(\frac{|y|^{\alpha}}{2\Delta}\right) \\ &\quad + \sum_{i=1}^{|y|-1} \left\{ \psi\left(\frac{i^{\alpha}}{2\Delta} + \frac{1}{2}\right) - \psi\left(\frac{i^{\alpha}}{2\Delta}\right) \right\}. \end{aligned} \quad (79) $$

For the first term in (79), we use (74) which implies that, as $|y| \to \infty$,

$$ \psi\left(\frac{1}{2}\right) - \psi\left(\frac{|y|^{\alpha}}{2\Delta}\right) = \log\left(\frac{\Delta}{|y|^{\alpha}}\right) + O(1) = -\alpha \log|y| + O(1). $$

For the second term, since $\frac{i^\alpha}{2\Delta} \to \infty$ as $i \to \infty$, it follows from (73) that as $i \to \infty$,

$$ \psi\left(\frac{i^{\alpha}}{2\Delta} + \frac{1}{2}\right) - \psi\left(\frac{i^{\alpha}}{2\Delta}\right) = \frac{\Delta}{i^{\alpha}}(1 + o(1)). $$

This implies that

$$ \sum_{i=1}^{|y|-1} \left\{ \psi\left(\frac{i^{\alpha}}{2\Delta} + \frac{1}{2}\right) - \psi\left(\frac{i^{\alpha}}{2\Delta}\right) \right\} \sim \sum_{i=1}^{|y|-1} \frac{\Delta}{i^{\alpha}} = O(1) $$

as $|y| \to \infty$.

□

### A.6 Variance of the potential when $\alpha > 1$

**Lemma A.5.** Assume $\alpha > 1$. Then for almost every $\omega \in \Omega$, as $|y| \to \infty$ we have that

$$ \sum_{O_n \prec x \prec y} \mathrm{Var}[\log(\rho_x)] = O(1). $$

*Proof.* It follows from (72) that for all $t \in [0, 1]$,

$$ \sum_{O_{\infty} \prec x \prec y} \mathrm{Var}(\log \rho_x) = \sum_{i=1}^{|y|} \left\{ \psi'\left(\frac{(i-1)^{\alpha}}{2\Delta} + \frac{1}{2}\right) + \psi'\left(\frac{i^{\alpha}}{2\Delta}\right) \right\}. $$

Since both arguments in the derivatives of the digamma function blow up as $i \to \infty$, it follows from (75) that as $i \to \infty$,

$$ \psi'\left(\frac{(i-1)^{\alpha}}{2\Delta} + \frac{1}{2}\right) + \psi'\left(\frac{i^{\alpha}}{2\Delta}\right) = \frac{4\Delta}{i^{\alpha}}(1 + o(1)). $$
This implies that

$$ \sum_{O_{\infty} < x \le y} \mathrm{Var}(\log \rho_x) \sim \sum_{i=1}^{|y|} \frac{4\Delta}{i^{\alpha}} = O(1) $$

as $|y| \to \infty$.

□

## References

[ACT19] J. Akahori, A. Collevecchio, and M. Takei. Phase transitions for edge-reinforced random walks on the half-line. *Electronic Communications in Probability*, 24, 2019.

[ADH13] R. Abraham, J-F. Delmas, and P. Hoscheit. A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. *Electron. J. Probab.*, 18(14):1–21, 2013.

[AEW13] S. Athreya, M. Eckhoff, and A. Winter. Brownian motion on $\mathbb{R}$-trees. *Transactions of the American Mathematical Society*, 365(6):3115–3150, 2013.

[Aid08] E. Aidékon. Transient random walks in random environment on a galton–watson tree. *Probability Theory and Related Fields*, 142:525–559, 2008.

[Ald91a] D. Aldous. Asymptotic fringe distributions for general families of random trees. *Ann. Appl. Probab.*, 1(2):228–266, 1991.

[Ald91b] D. Aldous. The continuum random tree. I. *Ann. Probab.*, 19(1):1–28, 1991.

[Ald91c] D. Aldous. The continuum random tree. II. An overview. In *Stochastic analysis (Durham, 1990)*, volume 167 of *London Math. Soc. Lecture Note Ser.*, pages 23–70. Cambridge Univ. Press, Cambridge, 1991.

[Ald93] D. Aldous. The continuum random tree. III. *Ann. Probab.*, 21(1):248–289, 1993.

[ALW16] S. Athreya, W. Löhr, and A. Winter. The gap between gromov-vague and gromov-hausdorff-vague topology. *Stochastic Processes and their Applications*, 126(9):2527–2553, 2016.

[ALW17] S. Athreya, W. Löhr, and A. Winter. Invariance principle for variable speed random walks on trees. *Ann. Probab.*, 45(2):625–667, 2017.

[And21] G. Andriopoulos. Invariance principles for random walks in random environment on trees. *Electron. J. Probab.*, 26:Paper No. 115, 38, 2021.

[Arc20a] E. Archer. Infinite stable looptrees. *Electronic Journal of Probability*, 25, 2020.

[Arc20b] E Archer. Random walks on decorated Galton-Watson trees, 2020.

[Bac11] S. Bacallado. Bayesian analysis of variable-order, reversible Markov chains. *Ann. Statist.*, 39(2):838–864, 2011.

[Bar98] M. Barlow. Diffusions on fractals. In *Lectures on probability theory and statistics (Saint-Flour, 1995)*, volume 1690 of *Lecture Notes in Math.*, pages 1–121. Springer, Berlin, 1998.

[BB99] R. Bass and K. Burdzy. Stochastic bifurcation models. *Ann. Probab.*, 27(1):50–108, 1999.

[BBI01] D. Burago, Y. Burago, and S. Ivanov. *A course in metric geometry*, volume 33 of *Graduate Studies in Mathematics*. American Mathematical Society, Providence, RI, 2001.

[BCP09] S. Bacallado, J. Chodera, and V. Pande. Bayesian comparison of markov models of molecular dynamics with detailed balance constraint. *The Journal of chemical physics*, 131(4):07B622, 2009.
[Ber92] J. Bertoin. An extension of Pitman's theorem for spectrally positive Lévy processes. *Ann. Probab.*, 20(3):1464-1483, 1992.

[Ber96] J. Bertoin. *Lévy processes*, volume 121 of *Cambridge Tracts in Mathematics*. Cambridge University Press, Cambridge, 1996.

[BGT89] N. Bingham, C. Goldie, and J. Teugels. *Regular variation*, volume 27 of *Encyclopedia of Mathematics and its Applications*. Cambridge University Press, Cambridge, 1989.

[Bil99] P. Billingsley. *Convergence of probability measures*. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999. A Wiley-Interscience Publication.

[BK06] M. Barlow and T. Kumagai. Random walk on the incipient infinite cluster on trees. *Illinois J. Math.*, 50(1-4):33-65, 2006.

[BPFT16] S. Bacallado, V. Pande, S. Favaro, and L. Trippa. Bayesian regularization of the length of memory in reversible sequences. *J. R. Stat. Soc. Ser. B. Stat. Methodol.*, 78(4):933-946, 2016.

[Bro86] T. Brox. A one-dimensional diffusion process in a Wiener medium. *Ann. Probab.*, 14(4):1206-1218, 1986.

[BST21] S. Bacallado, C. Sabot, and P. Tarrès. The*-edge-reinforced random walk. *arXiv preprint arXiv:2102.08984*, 2021.

[Car97] P. Carmona. The mean velocity of a brownian motion in a random lévy potential. *The Annals of Probability*, pages 1774-1788, 1997.

[CD86] D. Coppersmith and P. Diaconis. Random walk with reinforcement. *Unpublished manuscript*, 1986.

[CK08] D. Croydon and T. Kumagai. Random walks on galton-watson trees with infinite variance off-spring distribution conditioned to survive. *Electronic Journal of Probability*, 13:1419-1441, 2008.

[CK14] N. Curien and I. Kortchemski. Random stable looptrees. *Electron. J. Probab.*, 19(108):1-35, 2014.

[Col06] A. Collevecchio. Limit theorems for reinforced random walks on certain trees. *Probab. Theory Related Fields*, 136(1):81-101, 2006.

[Cro17] D. Croydon. An introduction to stochastic processes associated with resistance forms and their scaling limits. *RIMS Kokyuroku 2030, no. 1*, 2017.

[Cro18] D. Croydon. Scaling limits of stochastic processes associated with resistance forms. *Ann. Inst. Henri Poincaré Probab. Stat.*, 54(4):1939-1968, 2018.

[DF80] P. Diaconis and D. Freedman. de finetti's theorem for markov chains. *The Annals of Probability*, pages 115-130, 1980.

[DLG02] T. Duquesne and J-F. Le Gall. Random trees, Lévy processes and spatial branching processes. *Astérisque*, (281):vi+147, 2002.

[DLG05] T. Duquesne and J-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. *Probab. Theory Related Fields*, 131(4):553-603, 2005.

[DLG06] T. Duquesne and J-F. Le Gall. The Hausdorff measure of stable trees. *ALEA Lat. Am. J. Probab. Math. Stat.*, 1:393-415, 2006.

[DR06] P. Diaconis and S. Rolles. Bayesian analysis for reversible Markov chains. *Ann. Statist.*, 34(3):1270-1292, 2006.

[Duq03a] T. Duquesne. A limit theorem for the contour process of conditioned Galton-Watson trees. *Ann. Probab.*, 31(2):996-1027, 2003.
[Duq03b] T. Duquesne. A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab., 31(2):996-1027, 2003.

[Duq09] T. Duquesne. Continuum random trees and branching processes with immigration. Stochastic Process. Appl., 119(1):99-129, 2009.

[DV02] B. Davis and S. Volkov. Continuous time vertex-reinforced jump processes. Probability theory and related fields, 123(2):281-300, 2002.

[DW11] T. Duquesne and G. Wang. Exceptionally small balls in stable trees. arXiv preprint arXiv:1111.3465, 2011.

[EPW06] S. Evans, J. Pitman, and A. Winter. Rayleigh processes, real trees, and root growth with regrafting. Probab. Theory Related Fields, 134(1):81-126, 2006.

[GH10] C. Goldschmidt and B. Haas. Behavior near the extinction time in self-similar fragmentations. I. The stable case. Ann. Inst. Henri Poincaré Probab. Stat., 46(2):338-368, 2010.

[HH14] P. Hall and C. Heyde. Martingale limit theory and its application. Academic press, 2014.

[JM05] S. Janson and J-F. Marckert. Convergence of discrete snakes. J. Theoret. Probab., 18(3):615-647, 2005.

[Kes86a] H. Kesten. The limit distribution of Sinaï's random walk in random environment. Phys. A, 138(1-2):299-309, 1986.

[Kes86b] H. Kesten. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist., 22(4):425-487, 1986.

[Kor17] I. Kortchemski. Sub-exponential tail bounds for conditioned stable Bienaymé-Galton-Watson trees. Probability Theory and Related Fields, 168(1):1-40, 2017.

[KR19] I. Kortchemski and L. Richier. Condensation in critical Cauchy Bienaymé-Galton-Watson trees. The Annals of Applied Probability, 29(3):1837-1877, 2019.

[Kum04] T. Kumagai. Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms. Publ. Res. Inst. Math. Sci., 40(3):793-818, 2004.

[LG06] J-F. Le Gall. Random real trees. Ann. Fac. Sci. Toulouse Math. (6), 15(1):35-62, 2006.

[LGLJ98] J-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: the exploration process. Ann. Probab., 26(1):213-252, 1998.

[LPW17] D. Levin, Y. Peres, and E. Wilmer. Markov chains and mixing times. American Mathematical Society, Providence, RI, 2017. Second edition of [MR2466937], With contributions by Elizabeth L. Wilmer, With a chapter on "Coupling from the past" by James G. Propp and David B. Wilson.

[LST20] T. Lupu, C. Sabot, and P. Tarrès. Fine mesh limit of the vrjp in dimension one and bass-burdzy flow. Probability Theory and Related Fields, 177(1):55-90, 2020.

[Mar20] C. Marzouk. Scaling limits of discrete snakes with stable branching. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 56(1):502-523, 2020.

[Mat95] Pertti Mattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability.

[MR07] F. Merkl and S. Rolles. A random environment for linearly edge-reinforced random walks on infinite graphs. Probability theory and related fields, 138(1-2):157-176, 2007.

[Nev86] J. Neveu. Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist., 22(2):199-207, 1986.
[Pac16] C. Pacheco. From the sinai's walk to the brox diffusion using bilinear forms. *arXiv preprint arXiv:1605.02826*, 2016.

[Pem88] R. Pemantle. Phase transition in reinforced random walk and RWRE on trees. *Ann. Probab.*, 16(3):1229-1241, 1988.

[Pit06] J. Pitman. *Combinatorial stochastic processes: Ecole d'eté de probabilités de saint-flour xxxii-2002*. Springer, 2006.

[Sei00] P. Seignourel. Discrete schemes for processes in random media. *Probability theory and related fields*, 118(3):293-322, 2000.

[Sin82] Y. Sinaï. The limit behavior of a one-dimensional random walk in a random environment. *Teor. Veroyatnost. i Primenen.*, 27(2):247-258, 1982.

[Sla68] R. Slack. A branching process with mean one and possibly infinite variance. *Z. Wahrscheinlichkeitstheorie und Verw. Gebiete*, 9:139-145, 1968.

[ST15] C. Sabot and P. Tarrès. Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. *J. Eur. Math. Soc. (JEMS)*, 17(9):2353-2378, 2015.

[ST17] C. Sabot and L. Tournier. Random walks in Dirichlet environment: an overview. *Ann. Fac. Sci. Toulouse Math. (6)*, 26(2):463-509, 2017.

[Tak00] M. Takeshima. Behavior of 1-dimensional reinforced random walk. *Osaka Journal of Mathematics*, 37(2):355-372, 2000.

[Tak21] M. Takei. Almost sure behavior of linearly edge-reinforced random walks on the half-line. *Electron. J. Probab.*, 26:Paper No. 104, 18, 2021.

[Tet91] P. Tetali. Random walks and the effective resistance of networks. *J. Theoret. Probab.*, 4(1):101-109, 1991.

[vdVW96] A. van der Vaart and J. Wellner. *Weak convergence and empirical processes*. Springer Series in Statistics. Springer-Verlag, New York, 1996. With applications to statistics.