  We prove a new result on the spectral gap and mixing time of a Markov chain
with Glauber dynamics on the space of Dyck paths (i.e., Catalan paths) and
their generalization, which we call colored Dyck paths. The proof uses the
comparison theorem of Diaconis and Saloff-Coste and our previous results.
  Let $2n$ be the number of spins. We prove that the gap of the Fredkin quantum
spin chain Hamiltonian [6, 20], is $\Theta(n^{-c})$ with $c\ge2$. Our results
on the spectral gap of the Markov chain are used to prove a lower bound of
$O(n^{-15/2})$ on the energy of first excited state above the ground state of
the Fredkin quantum spin chain. We prove an upper bound of $O(n^{-2})$ using
the universality of Brownian motion and convergence of Dyck random walks to
Brownian excursions. Lastly, the 'unbalanced' ground state energies are proved
to be polynomially small in $n$ by mapping the Hamiltonian to an effective
hopping Hamiltonian with next nearest neighbor interactions and analytically
solving its ground state.
