derive conductance bounds

This work addresses rapid mixing of Markov Chain Monte Carlo (MCMC) algorithms on high-dimensional discrete distributions—such as vertex colorings of graphs and bases of matroids. We introduce a novel analytical framework centered on **spectral independence**, unifying spectral independence, the local-to-global theorem, and Oppenheim’s trickle-down theorem for the first time. This yields a cohesive spectral analysis paradigm applicable to both graphical models and matroid structures. We prove that spectral independence implies polynomial mixing time; under natural conditions, it yields the optimal $O(n log n)$ mixing time for Glauber dynamics and tight spectral gap bounds for base-exchange walks. Our results systematically extend Anari et al.’s matroid sampling framework, providing universal, tight convergence guarantees for probabilistic inference and randomized algorithms over combinatorial structures.

This paper uncovers a fundamental equivalence between effective resistance on graphs and optimal transport problems, addressing the computational intractability of Wasserstein distances on graphs. Method: We systematically construct the p-Beckmann distance family and establish the first rigorous mathematical bridge between effective resistance and optimal transport. Our variational framework—based on flow constraints—derives a Benamou–Brenier-type formulation for the 2-Beckmann distance and reveals deep connections to random walks, optimal stopping times, and graph Sobolev spaces. Contribution/Results: (1) We unify metric foundations across graph combinatorics, discrete geometry, and machine learning; (2) we propose a scalable, geometrically faithful alternative to Wasserstein distances; (3) our approach significantly alleviates computational bottlenecks inherent in graph Wasserstein computation—preserving expressive power while improving efficiency—thereby enabling a new paradigm for unsupervised learning on graph-structured data.

This study addresses the problem of comparing convergence efficiency for reversible Markov chains. Methodologically, it unifies three fundamental metrics—spectral gap, asymptotic variance of ergodic averages, and graph-theoretic connectivity—within a single variational framework. It establishes, for the first time, rigorous variational equivalence among derivative-type Dirichlet forms, asymptotic variance, and connectivity, and proves that nonzero connectivity is equivalent to the existence of a nonzero right spectral gap. The analysis integrates variational methods, spectral graph theory, ergodic theory, and functional inequalities. The resulting framework provides the first systematic and computationally tractable variational tool for convergence analysis of reversible Markov chains. It significantly enhances both the theoretical precision of efficiency assessment and the capability for optimization-driven design of MCMC algorithms.

This work resolves a long-standing open problem concerning the mixing time of Gibbs sampling at the computational phase transition point—the tree uniqueness threshold—for hardcore and Ising models. Specifically, it establishes the first rigorous polynomial upper and lower bounds on the mixing time of Glauber dynamics at criticality: for the hardcore model with fugacity λ = λ_c(Δ) and the Ising model with inverse temperature β = β_c on graphs of maximum degree Δ ≥ 3. The approach introduces a novel framework integrating field dynamics, proximal samplers, and localization analysis (Chen–Eldan framework), combined with spectral independence and high-dimensional probability tools. Results show that the hardcore model mixes in Õ(n^{2+4/e})–Ω(n^{4/3}) steps, while the Ising model mixes in Õ(n²)–Ω(n^{3/2}) steps at critical temperature, and achieves a tight bound of Õ(n^{3/2}) under the critical interaction matrix. This fills a fundamental gap in computational phase transition theory regarding convergence rates precisely at criticality.

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 problem of characterizing graph structures with constant mixing time from a purely combinatorial perspective—a longstanding gap, as classical spectral graph theory only yields logarithmic mixing-time bounds. We propose a novel combinatorial criterion based on *bipartite density of small sets*, which is strictly weaker than near-optimal spectral radius constraints yet strictly stronger than small-set vertex expansion. Our method integrates spectral analysis, random walk theory, and extremal combinatorics to establish a necessary and sufficient condition linking small-set bipartite density directly to constant mixing time. We rigorously prove this equivalence and validate its effectiveness across several canonical graph families, including expanders, random regular graphs, and certain product graphs. This result provides the first combinatorial characterization equivalent to constant mixing time, bridging a fundamental theoretical gap in mixing-time analysis. It establishes a new structural paradigm for both analyzing and constructing rapidly mixing graphs, with implications for algorithm design, sampling, and distributed computation.

This work resolves a long-standing open problem concerning whether the lazy swap Markov chain on binary matrices with prescribed row and column sums exhibits a polynomial spectral gap—a question central to the rapid mixing of null-model samplers in ecology, statistics, and related fields. By reducing the general $m \times n$ case to the three-row setting and combining a comparison with a two-row heat-bath chain, analysis of column-count variables, and a harmonic decomposition over Johnson graphs, the authors establish—for the first time—a tight inverse-polynomial lower bound on the spectral gap of $\Omega\big((\binom{m}{2}\binom{n}{2})^{-1}\big)$ for all feasible margins. This result confirms the rapid mixing conjecture of Kannan–Tetali–Vempala (1997) and establishes that the lazy swap chain mixes rapidly under any feasible marginal constraints.

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 $\Theta(n^{3/2})$.

This work addresses a critical limitation of Laplacian spectral sparsification: while it preserves quadratic forms, it offers little control over the adjacency spectral radius—the largest eigenvalue of the adjacency matrix—which plays a pivotal role in NIMFA epidemic thresholds and spectral clustering. The paper establishes, for the first time, tight error bounds on the adjacency spectral radius under Laplacian sparsification. It introduces a novel analytical framework combining Perron–Frobenius monotonicity, eigenvector delocalization, and resolvent perturbation theory, integrated with effective resistance sampling and Matrix Bernstein inequalities. The resulting deterministic bound is |λ₁(A_H) − λ₁(A_G)| ≤ ε(2Δ − λ₁), and a high-probability bound of O(εΔ/√c). For Erdős–Rényi graphs, regular expanders, and stochastic block models, refined bounds of O(εΔ√(log n)/√n + ε²Δ²/δ_gap) are derived, with tightness proven for regular graphs.