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We prove a new result on the spectral gap and mixing time of a Markov chainwith Glauber dynamics on the space of Dyck paths (i.e., Catalan paths) andtheir generalization, which we call colored Dyck paths. The proof uses thecomparison 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 quantumspin chain Hamiltonian [6, 20], is $\Theta(n^{-c})$ with $c\ge2$. Our resultson 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 ofthe Fredkin quantum spin chain. We prove an upper bound of $O(n^{-2})$ usingthe universality of Brownian motion and convergence of Dyck random walks toBrownian excursions. Lastly, the 'unbalanced' ground state energies are provedto be polynomially small in $n$ by mapping the Hamiltonian to an effectivehopping Hamiltonian with next nearest neighbor interactions and analyticallysolving its ground state.

The gap of Fredkin quantum spin chain is polynomially small