2 corrected a typo

Let $T$ be a rooted Galton-Watson random tree generated accordingly to a probability distribution $\mu$. Now assign to each edge $e$ a random non-negative weight $w_e$ distributed a accordingly to a distribution $\nu$. We also assume that the weights are independent for different edges.

Let $T_{n}$ be the collection of nodes at (hop) distance $n$ from the root. For each $v\in T_{n}$, let $P_{v}$ denote the collection of paths path from the root to $v$. Define

$$Z_{v}=\min_{v \in P_{v}}\sum_{e\in p}{w_e}Z_{v}=\sum_{e\,\in P_{v}}{\,w_e}.$$

Now for each $n$ let $Y_{n}=\min_{v\in T_{n}}Z_{v}$. It was proved in Limit distributions for minimal displacement of branching random walks that the sequence of random variables $$\{Y_{n}-\mathbb{E}(Y_{n})\}_{n\geq 1}$$ is tight.

My question are:

• Is it known what is the behavior of $\mathbb{E}(Y_{n})$ as $n$ increases in terms of $\mu$ and $\nu$?

• Is it known for the case $\mu=\delta_{k}$, i.e. when $T$ is a $k+1$ regular tree?

1

# First Passage Percolation on Trees

Let $T$ be a rooted Galton-Watson random tree generated accordingly to a probability distribution $\mu$. Now assign to each edge $e$ a random non-negative weight $w_e$ distributed a accordingly to a distribution $\nu$. We also assume that the weights are independent for different edges.

Let $T_{n}$ be the collection of nodes at (hop) distance $n$ from the root. For each $v\in T_{n}$, let $P_{v}$ denote the collection of paths from the root to $v$. Define

$$Z_{v}=\min_{v \in P_{v}}\sum_{e\in p}{w_e}.$$

Now for each $n$ let $Y_{n}=\min_{v\in T_{n}}Z_{v}$. It was proved in Limit distributions for minimal displacement of branching random walks that the sequence of random variables $$\{Y_{n}-\mathbb{E}(Y_{n})\}_{n\geq 1}$$ is tight.

My question are:

• Is it known what is the behavior of $\mathbb{E}(Y_{n})$ as $n$ increases in terms of $\mu$ and $\nu$?

• Is it known for the case $\mu=\delta_{k}$, i.e. when $T$ is a $k+1$ regular tree?