I am researching closed random walks on graphs and have the following problem that I haven't been able to find a reference for. 


Consider a random walk on $\mathbb Z$ starting at 0 and at each step it moves $-1$ or $+1$ each with probability $1/2$. If the walk has length $2n$ it is well-known that the support (or how many elements of $\mathbb Z$ that are covered by the walk) is $\Theta(\sqrt n)$ with high probability. 


Suppose now that our walk is closed, i.e. we condition on that the walk starts and ends at 0. Is it still the case that the walk has support $\Theta(\sqrt n)$ with high probability?


I would be happy if I could just show that the support is at least $\Omega(n^{\varepsilon})$ for some constant $\varepsilon>0$.



Support of closed random walk on $\mathbb Z$
One way to do this is as follows. We have to show that 
$$P(M\_n\ge x|S\_n=0)\to1$$
(as $n\to\infty$) if $x=o(\sqrt n)$, where $S\_n$ is the position of the walk at time $n$ and $M\_n:=\max\_{0\le k\le n}S\_k$. By the reflection principle (see e.g. [Theorem 0.8](http://cgm.cs.mcgill.ca/~breed/MATH671/lecture2corrected.pdf)) and the [de Moivre--Laplace theorem](https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem) , for natural $x$ such that $x=o(\sqrt n)$,
$$P(M\_n\ge x,S\_n=0)=P(S\_n=2x)\sim P(S\_n=0),$$
whence
$$P(M\_n\ge x|S\_n=0)=\frac{P(M\_n\ge x,S\_n=0)}{P(S\_n=0)}\to1,$$
as desired. 


