mixing time analysis

Non-asymptotic mixing-time analysis of finite-state ergodic Markov chains—both reversible and irreversible—remains challenging, particularly due to the lack of a unified convergence characterization for irreversible chains. Method: We develop an operator-theoretic framework based on orthogonal projections of the transition operator in the ℓ²(π) space, using matrix norms to quantify convergence rates. Contribution/Results: We establish, for the first time, submultiplicativity of pointwise χ²-divergence for irreversible chains, yielding explicit, computable bounds dependent on spectral structure, algebraic–geometric multiplicity gaps, and condition numbers of similarity transformations. We apply this framework to momentum-based samplers, revealing their diffusive behavior via hypercontractivity and regression analysis. For irreversible triangular random walks on graphs, we derive the tightest known mixing-time bound to date and demonstrate near-optimal O(n¹·⁹⁷) convergence even under V-shaped target distributions.

This work addresses the quantification of L² relaxation times for irreversible Markov processes by establishing a unified analytical framework based on the singular value gap of the generator and its hypoelliptic structure. The approach applies seamlessly to continuous-time Markov chains, diffusions, and piecewise deterministic processes, encompassing both first- and second-order hypoelliptic settings under degenerate noise. It extends and simplifies the theory of hypocoercivity, yielding sharp upper and lower bounds on relaxation times. As notable applications, the method confirms the conjecture of Diaconis and Miclo that lifted random walks on Abelian groups achieve a square-root speedup, and provides precise convergence rate estimates for noisy switching flows and irreversible diffusions.

This work establishes Rosenthal- and Bernstein-type concentration inequalities for additive functionals of geometrically ergodic Markov chains, explicitly characterizing the dependence of deviation bounds on mixing time. Methodologically, it pioneers the extension of the classical Rosenthal inequality to the Markov-dependent setting via a novel analytical framework based on Poisson equation decomposition, which precisely links mixing constants, martingale Rosenthal constants, and deviation bounds. Integrating martingale techniques, geometric ergodicity analysis, and quantitative mixing time estimation, the approach yields computable, explicit, and tight upper bounds—significantly improving the polynomial dependence on mixing time present in prior results. The derived inequalities provide a rigorous theoretical foundation for error control in MCMC algorithms, sequential Monte Carlo estimation, and large-sample inference for non-i.i.d. statistics.

This paper addresses the estimation of β-mixing coefficients for geometrically ergodic Markov processes, aiming to quantify temporal dependence from a single observed trajectory. We propose a nonparametric estimation framework grounded in kernel density estimation and statistical learning theory, integrating tools from Besov space analysis, mixing process theory, and concentration inequalities. Our main contributions are threefold: (1) Under Besov-type regularity assumptions on the stationary density, we establish the first convergence rate with an explicit logarithmic correction factor; (2) For finite-state-space Markov chains, we achieve the optimal expected error rate $O(log n / sqrt{n})$ and corresponding high-probability bounds—without requiring any density boundedness or smoothness assumptions; (3) In continuous state spaces, we derive an expected error rate of $O(log n cdot n^{-s/(2s+2)})$, where $s > 0$ denotes the Besov smoothness index, along with matching high-probability guarantees.

This paper addresses the design of fastest-mixing reversible Markov chains: given a target stationary distribution, how to construct a reversible chain with minimal mixing time. Focusing on two fundamental graph structures—clique-lifted graphs and arbitrary subgraphs—the work makes two key contributions: (1) It establishes, for the first time, a strict reduction of the fastest-mixing problem on clique-lifted graphs to that on their base graphs, preserving the optimal mixing time; (2) It proves that the optimal transition probabilities on a subgraph can be determined independently of the ambient graph, ensuring consistency between local optimization and global optimality. Methodologically, the approach integrates semidefinite programming (SDP), graph-theoretic techniques (clique lifting and subgraph embedding), and spectral analysis of reversible chains. Experiments demonstrate that the framework yields analytically optimal solutions across diverse topologies, combining theoretical rigor with computational scalability.

This work addresses the challenge of analyzing mixing times for Markov chain Monte Carlo (MCMC) algorithms under non-convex potentials and heavy-tailed target distributions by introducing a unified contraction framework based on $\mathsf{E}_\gamma$-divergence. The framework incorporates Gaussian smoothing to derive an explicit global contraction coefficient and introduces a local contraction coefficient to handle unbounded importance weights, thereby effectively accommodating heavy-tailed settings. It applies broadly to algorithms such as projected Langevin Monte Carlo and independent Metropolis–Hastings. For the former, it establishes, for the first time, explicit exponential convergence rates under KL, $\chi^2$, and Rényi divergences; for the latter, it recovers or improves existing convergence bounds under either finite-moment or heavy-tailed conditions.

This work investigates the rapid mixing of Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. By ingeniously adapting the Bochner–Bakry–Émery method—originally developed for continuous-space Gibbs point processes—to the discrete setting, the authors directly establish a Poincaré inequality. Their approach leverages an expansion of the Dirichlet form and eliminates squared-sum terms, thereby circumventing intricate recursive arguments or coupling constructions. This yields a concise, self-contained proof of rapid mixing that requires no additional assumptions, significantly streamlining the existing theoretical framework. Notably, it provides the first explicit lower bound on the spectral gap in the non-uniqueness regime.

This work addresses the long-standing gap between the best-known upper bound of $\widetilde{O}(n^3)$ and the theoretical lower bound of $\Omega(n^{3/2})$ for the mixing time of the flip Markov chain on triangulations of a convex $(n+2)$-gon. By developing a novel analysis framework based on transport flows, combined with combinatorial decomposition and functional inequality techniques, the authors significantly refine the characterization of the chain’s convergence rate. This approach yields improved upper bounds of $\widetilde{O}(n^2)$ for both the relaxation time and the logarithmic Sobolev constant, thereby tightening the mixing time bound from $\widetilde{O}(n^3)$ to $\widetilde{O}(n^2)$. The result represents a substantial step toward resolving Aldous’s conjecture that the true mixing time is $\Theta(n^{3/2})$.