We introduce a unified operator-theoretic framework for analyzing mixing times of finite-state ergodic Markov chains that applies to both reversible and non-reversible dynamics. The central object in our analysis is the projected transition operator $PU_{ \perp 1}$, where $P$ is the transition kernel and $U_{ \perp 1}$ is orthogonal projection onto mean-zero subspace in $\ell^{ 2}(\pi)$, where $\pi$ is the stationary distribution. We show that explicitly computable matrix norms of $(PU_{ \perp 1})^k$ gives non-asymptotic mixing times/distance to stationarity, and bound autocorrelations at lag $k$. We establish, for the first time, submultiplicativity of pointwise chi-squared divergence in the general non-reversible case. We provide for all times $\chi^{ 2}(k)$ bounds based on the spectrum of $PU_{ \perp 1}$, i.e., magnitude of its distinct non-zero eigenvalues, discrepancy between their algebraic and geometric multiplicities, condition number of a similarity transform, and constant coming from smallest atom of stationary distribution(all scientifically computable). Furthermore, for diagonalizable $PU_{ \perp 1}$, we provide explict constants satisfying hypocoercivity phenomenon for discrete time Markov Chains. Our framework enables direct computation of convergence bounds for challenging non-reversible chains, including momentum-based samplers for V-shaped distributions. We provide the sharpest known bounds for non-reversible walk on triangle. Our results combined with simple regression reveals a fundamental insight into momentum samplers: although for uniform distributions, $n\log{ n}$ iterations suffice for $\chi^{ 2}$ mixing, for V-shaped distributions they remain diffusive as $n^{ 1.969}\log{ n^{ 1.956}}$ iterations are sufficient. The framework shows that for ergodic chains relaxation times $\tau_{ rel}=\|\sum_{ k=0}^{ \infty}P^{ k}U_{ \perp 1}\|_{ \ell^{ 2}(\pi)}$.
Categories: math.PR stat.CO