Let $H_N=\{0,1\}^N$ the N-dimensional hypercube. We define the following random walk $X_n$ on $H_N$:

- start from a point $x \in H_N$
- pick at random an integer $k$ in $[1,N-1]$ and exchange $x(k)$ and $x(k+1)$
- go on like that...

For instance, with n=4, the random walk looks like: $X_0=0011$, $X_1=0101$, $x_2=0110$, ...

Do you know any article/textbook where this random walk is studied in details ? I have been looking in many classical textbook without success...

Beyond this general request, I am interested in the following question: take a given point $x^* \in H_N$ and define $d_n$ the Hamming distance between $X_n$ and $x^*$. Is is possible to estimate the mean of the first time when $d_n$ hits $0$ (as a function of $x*$ and the starting point of the chain $X_0$) ? Of course we suppose that $X_0$ and $x^*$ have the same number of ones... I hope you will enjoy this problem.

Thank you !