We study approximately uniform sampling of satisfying assignments from random $k$-SAT formulas. Fix a constant $B>0$. For every $k\ge 2^{20}$ and density $0<α\le 2^k/k^{16}$, we prove that, with high probability over a random formula on $n$ variables, every accuracy $n^{-B}\le\varepsilon\le 1/2$ admits a sampler whose output distribution is within total variation distance $\varepsilon$ of the uniform distribution on satisfying assignments with expected running time $$\widetilde O_B\left(\mathrm{poly}(k,α+1)n^{2+η}\right),\quad 0<η\le 2^{16-k}.$$ In particular, the exponent of $n$ is $2+O(2^{-k})$ while the dependence on the width and density is polynomial with a universal degree. This improves the parameter-dependent polynomial exponent of the sampling algorithm of Chen, Lonkar, Wang, Yang, and Yin (STOC 2025) in the density regime $2^k/\mathrm{poly}(k)$. Our algorithm separates a high-degree core from the remaining variables. Every residual formula on the good variables is a deletion subformula of a common bounded-dependency formula. Adapting the rejection--repair framework of Liu, Wang, Yin, Zhang and Zhou (arXiv 2026), we construct an executable sampler for all such residual formulas using ordinary connected violation sets replacing weighted 2-tree witnesses and use it to simulate approximate block heat-bath updates on the core. To analyze the resulting core chain, we integrate out the good variables, represent the core marginal by an absolutely convergent polymer expansion and combine a component-swap variance comparison with an adaptive exploration bound. Absorbing single-component interactions into the local weights leaves interactions of cluster size at least two; retaining their additional decay gives a spectral gap of at least $1/(2s n^{2^{12-k}})$ where $s$ counts core components. Together with the residual sampler, this yields the stated running time.
We develop a framework for deterministic approximate counting of multi-spin systems beyond bounded-degree graphs. The algorithm recursively constructs rational polytopes containing the true marginal vectors and uses linear-fractional programming to obtain certified bounds on marginal ratios. For positive interactions on graphs of polynomial connective constant $D$, we establish strong spatial mixing whenever $Dc<1$, where $c$ bounds the Birkhoff contraction coefficients of the interactions. Under the same condition, we also give a deterministic fully polynomial-time approximation scheme (\textbf{FPTAS}) for the partition function. We further extend the framework to proper colorings of sparse Erdős-Rényi random graphs using recursion on permissive blocks. For every fixed $η\in(0,1)$, sufficiently large fixed $d$, and fixed integer $q\ge(2+η)d$, we obtain an \textbf{FPTAS} for counting proper $q$-colorings of $G\sim\mathcal G(n,d/n)$ with high probability over $G$. This improves the leading constant $3$ in the earlier counting guarantee of Yin and Zhang (APPROX/RANDOM, 2016) to $2$, and asymptotically matches the spatial mixing regime established by Yin (ICALP, 2014).
We study the mixing time of the single-site Glauber dynamics for sampling proper $q$-colorings on sparse Erdős-Rényi random graphs $\mathcal{G}(n, d/n)$. We prove that, whenever $q \ge (1 + o_d(1))d$, Glauber dynamics mixes in polynomial time with high probability over $\mathcal{G}(n, d/n)$ for all sufficiently large integers $n$. This improves the previous bound $q = (α^* + o_d(1))d$ where $α^* = 1.7632...$ is the unique solution of $x = \exp(1/x)$ on $[1, \infty)$ by Efthymiou, Hayes, Štefankovič, and Vigoda (SODA, 2018) and asymptotically matches the uniqueness threshold on infinite regular trees. Our proof has three main ingredients. First we establish logarithmic bounds on the spectral and coupling independence constants of the Gibbs distribution by extending the tree analysis of Chen, Liu, Mani, and Moitra (J. ACM, 2026). Second, we extend the entropy-factorization framework for two-spin systems on random graphs, developed by Bezáková, Galanis, Goldberg, and Štefankovič (TALG, 2024), to proper colorings by introducing a buffer around high-degree vertices. Finally, we compare the resulting heat-bath block dynamics with the single-site Glauber dynamics.