Spectral Gap and Mixing Time of the Lazy Walk on a Cycle

Consider the lazy random walk on the cycle graph $C_n$: at each step, the walker stays put with probability $\tfrac{1}{2}$, and moves to each of the two neighbors with probability $\tfrac{1}{4}$. The transition matrix has eigenvalues $\lambda_k = \tfrac{1}{2}(1 + \cos(2\pi k / n))$ for $k = 0, 1, \ldots, n-1$.

(a) Find the spectral gap $\gamma = 1 - \lambda_1$ in terms of $n$.

(b) Using the relation $t_{\text{mix}} \asymp \frac{1}{\gamma}$ (up to logarithmic factors), determine the order of the mixing time for the lazy walk on $C_n$ as $n \to \infty$.