Improved Bounds for Sampling Solutions of Random CNF Formulas

Let $\Phi$ be a random $k$-CNF formula on $n$ variables and $m$ clauses, where each clause is a disjunction of $k$ literals chosen independently and uniformly. Our goal is to sample an approximately uniform solution of $\Phi$ (or equivalently, approximate the partition function of $\Phi$). Let $\alpha=m/n$ be the density. The previous best algorithm runs in time $n^{\mathsf{poly}(k,\alpha)}$ for any $\alpha\lesssim2^{k/300}$ [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. Our result significantly improves both bounds by providing an almost-linear time sampler for any $\alpha\lesssim2^{k/3}$. The density $\alpha$ captures the \emph{average degree} in the random formula. In the worst-case model with bounded \emph{maximum degree}, current best efficient sampler works up to degree bound $2^{k/5}$ [He, Wang, and Yin, FOCS'22 and SODA'23], which is, for the first time, superseded by its average-case counterpart due to our $2^{k/3}$ bound. Our result is the first progress towards establishing the intuition that the solvability of the average-case model (random $k$-CNF formula with bounded average degree) is better than the worst-case model (standard $k$-CNF formula with bounded maximal degree) in terms of sampling solutions.Comment: 51 pages, all proofs added, and bounds slightly improve

Merging costs for the additive Marcus-Lushnikov process, and Union-Find algorithms

Starting with a monodisperse configuration with $n$ size-1 particles, an additive Marcus-Lushnikov process evolves until it reaches its final state (a unique particle with mass $n$). At each of the $n-1$ steps of its evolution, a merging cost is incurred, that depends on the sizes of the two particles involved, and on an independent random factor. This paper deals with the asymptotic behaviour of the cumulated costs up to the $k$th clustering, under various regimes for $(n,k)$, with applications to the study of Union--Find algorithms.Comment: 28 pages, 1 figur

Algebraic renormalisation of regularity structures

We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations. Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory

Coding Theory

Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances

Percolation in weight-dependent random connection models

We study a general class of inhomogeneous spatial random graphs, the weight-dependent random connection model. Vertices are given through a standard Poisson point process in Euclidean space and each vertex carries additionally an i.i.d weight. Edges are drawn in such a way that short edges and edges to large weight vertices are preferred. This allows in particular the study of models that combine long-range interactions and heavy-tailed degree distributions. The occurrence of long edges together with the hierarchy of the vertices coming from the weights typically leads to very well connected graphs. We identify a sharp phase transition where the existence of a subcritical percolation phase becomes possible. This transition depends on both, the power-law of the degree distribution and on the geometric model parameter, showing the significant effect of clustering on the graph’s topology. We further study the specifics of dimension one in parameter regimes where a subcritical phase exists. Natural examples that are contained in our framework are for instance the random connection model, the Poisson Boolean model, scale-free percolation and the agedependent random connection model. We use our results to characterize robustness of age-based spatial preferential attachment networks

A mathematical foundation for the use of cliques in the exploration of data with navigation graphs

Navigation graphs were introduced by Hurley and Oldford (2011) as a graph-theoretic framework for exploring data sets, particularly those with many variables. They allow the user to visualize one small subset of the variables and then proceed to another subset, which shares a few of the original variables, via a smooth transition. These graphs serve as both a high level overview of the dataset as well as a tool for a first-hand exploration of regions deemed interesting. This work examines the nature of cliques in navigation graphs, both in terms of type and magnitude, and speculates as to what their significance to the underlying dataset might be. The questions answered by this body of work were motivated by the belief that the presence of cliques in navigation graphs is a potential indicator for the existence of an interesting, possibly unanticipated, relationship among some of the variables. In this thesis we provide a detailed examination of cliques in navigation graphs, both in terms of type, size and number. The study of types of cliques informs us of the potential significance of highly connected structures to the underlying data and guides our approach for examining the possible clique sizes and counts. On the other hand, the prevalence of large clique sizes and counts is suggestive of an interesting, possibly unexpected, relationship between the variates in the data. To address the challenges surrounding the nature of cliques in navigation graphs, we develop a framework for the derivation of closed-form expressions for the moments of count random variables in terms of their underlying indecomposable summands is established. We use this framework in conjunction with a connection between intersecting set families to obtain edge counts within a clique cover and thus, obtain closed-form expressions for the moments of clique counts in random graphs

Equivariant Neural Operator Learning with Graphon Convolution

We propose a general architecture that combines the coefficient learning scheme with a residual operator layer for learning mappings between continuous functions in the 3D Euclidean space. Our proposed model is guaranteed to achieve SE(3)-equivariance by design. From the graph spectrum view, our method can be interpreted as convolution on graphons (dense graphs with infinitely many nodes), which we term InfGCN. By leveraging both the continuous graphon structure and the discrete graph structure of the input data, our model can effectively capture the geometric information while preserving equivariance. Through extensive experiments on large-scale electron density datasets, we observed that our model significantly outperformed the current state-of-the-art architectures. Multiple ablation studies were also carried out to demonstrate the effectiveness of the proposed architecture

Smooth Edgeworth Expansion and Wasserstein-pp Bounds for Mixing Random Fields

The Edgeworth expansion is a central tool of probability that offers some refinement on the central limit theorem by providing higher-order approximations. In this paper, we consider $d$-dimensional mixing random fields $\bigl(X^{(n)}_{i}\bigr)_{i\in T_{n}}$ and study the empirical average $W_n:=\sigma_n^{-1} \sum_{i\in T_n}X^{(n)}_{i}$. Firstly, under mixing and moment conditions, we obtain a smooth Edgeworth expansion for $W_n$ to any order $p$. The proof relies on the Stein's method and a new constructive graph approach. Then we utilize the obtained expansion and a newly proposed method in Liu and Austern [2023] to obtain the first rates for the central limit theorem in Wasserstein-$p$ distance for arbitrary $p\ge 1$. Finally, we apply those results to obtain tail bounds and non-uniform Berry-Esseen bounds with polynomial decay.Comment: 92 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2209.0937