# Random walks on complete binary trees

Let $$T$$ to be a complete binary tree of height $$n$$ and root $$r$$.

A random walk begins at $$r$$, and at each step uniformly at random moves on a neighbor.

There is $$m$$ Random hikers from $$r$$ and let denote with $$H_1, dots, H_m$$, the heights reached by walkers after $$n$$ not.

Show that for a constant $$C$$ which should not depend on $$n$$
and $$m$$, he argues that

$$mathbb {P} ( underet {i in [m]} max left | H_i – frac {n} {3} right | ge C sqrt { n n}} ge 1 – frac {1} {n}$$

I've tried several strategies to define correctly $$H_i$$ as a sum of random variables and the like, but no one has turned out to work. Do you have an idea / suggestion to attach this problem?