  We prove tight bounds on the relaxation time of the so-called $L$-reversal
chain, which was introduced by R. Durrett as a stochastic model for the
evolution of chromosome chains. The process is described as follows. We have
$n$ distinct letters on the vertices of the ${n}$-cycle (${{\mathbb{Z}}}$ mod
$n$); at each step, a connected subset of the graph is chosen uniformly at
random among all those of length at most $L$, and the current permutation is
shuffled by reversing the order of the letters over that subset. We show that
the relaxation time $\tau (n,L)$, defined as the inverse of the spectral gap of
the associated Markov generator, satisfies $\tau (n,L)=O(n\vee
\frac{n^3}{L^3})$. Our results can be interpreted as strong evidence for a
conjecture of R. Durrett predicting a similar behavior for the mixing time of
the chain.
