probabilistic modelers

Abstract: We study approximately uniform sampling of satisfying assignments from random $k$-SAT formulas. For every sufficiently large $k$ and density $0 < α\le 2^k/k^{16}$, we prove that, with high probability over the formula, there is a sampler whose output distribution is within total variation distance $\varepsilon$ of the uniform distribution on satisfying assignments and whose expected running time is at most $(nk(α+1)/\varepsilon)^C$, for a universal constant $C$. Our algorithm improves the counting and sampling algorithms obtained by Chen, Lonkar, Wang, Yang, and Yin (STOC 2025) at the density $2^k/\operatorname{poly}(k)$ with running time $(n/\varepsilon)^{\operatorname{poly}(k,α)}$. Our result achieves this density region for sampling with a polynomial degree independent of both the width and the density. Our algorithm separates a high-degree core from the remaining variables, and combines a recursive sampler for the residual formulas with approximate block heat-bath updates on the core. We adapt the recursive insertion-chain framework of Jain, Mizgerd, and Pham (2026) from $2$-trees to ordinary connected violation sets. Expansion and random literal signs yield uniform moment bounds for the resulting correlated lists across all residual formulas, allowing the signed-flow analysis to give a universal polynomial running-time degree. A polymer expansion and an exploration bound establish a polynomial spectral gap for the core dynamics.

Abstract: We study parallel samplers that reveal each coordinate once and draw the coordinates of each batch independently from their exact conditional marginals. Under the same sampling interface and a hard three-round cap, randomization can improve accuracy beyond every deterministic policy that adapts its batches to previously observed values. For an explicit full-support binary family, the optimal deterministic adaptive forward Kullback-Leibler divergence grows linearly with the dimension, while a uniform mixture of four fixed schedules has exponentially small divergence. Only two independent fair bits are needed to select the schedule. We determine the asymptotic additive constant in the deterministic optimum and show that its optimal total variation tends to one quarter. An outward-rounded finite-sum certificate establishes divergence separation at dimension 389. The mechanism combines an irreversible first-batch dependence loss with posterior concentration in the randomized components. A density-coverage lemma describes how their mixture restores the target. A complementary noisy-matching family exhibits a growing round gap for nonadaptive schedules, although uniform random coordinate order still has linear mixture error. These results concern known finite structures and exact conditional marginals.

Abstract: We study the problem of computing class probabilities in block-independent disjoint (BID) probabilistic databases. Given the probability with which each block in the database realizes each feasible tuple type, the goal is to compute the probability of a class of worlds specified by a given tuple multiplicity vector, thus grouping together worlds with the same bag (multiset) of realized tuple types. For this problem, we prove $\#\mathsf{P}$-hardness even for very restricted and structured inputs. On the other hand, we show that it admits an FPRAS, as well as $\mathsf{XP}$-time algorithms parameterized by the number of tuple types and the treewidth of an incidence graph modeling the connections between blocks and tuples. Finally, we show that augmenting the problem with certain compatibility constraints between block realizations renders it $\#\mathsf{XLP}$- and $\#\mathsf{XALP}$-hard parameterized by pathwidth and treewidth respectively, ruling out $\mathsf{FPT}$ algorithms under standard assumptions. We leave as an open question whether this also holds in the absence of compatibility constraints.

Abstract: We introduce a new family of trickledown theorems, a.k.a., local to global technique to bound the spectral gap of the Glauber dynamics for multi-state spin systems. In this technique instead of upper-bounding the influence matrix of a link of co-dimension 2 by $λI$ (where $λ$ is the second eigenvalue of the link), we upper-bound the influence matrix after a carefully chosen rank-1 shift. The rank-1 shift allows for a significantly smaller upper-bound but it comes at the cost of bounding the average loss due to rank-1 perturbations. As an application we use this method to show that the natural Glauber dynamics mixes in polynomial time to generate samples from the Sherrington-Kirkpatrick model for $β\leq \tfrac{1}{2}+\varepsilon$, for an absolute constant $\varepsilon>0$. At the heart of the proof we manage to bound the loss due to rank-1 perturbations by averaging over all links of co-dimension 2.