arXiv subjects

Nathan Tung

Publications and source records attributed to Nathan Tung.

8 recordsLinked to original sources

arXiv2026

On efficient graph covers and steered random walks

We prove that the vertices of any $n$-vertex graph can be partitioned into pieces of radius $r = O(\log n)$ such that the sum of the sizes of their closed neighborhoods is at most $4n$. This answers a recent question of Bukh and Dubroff and directly yields an improvement to their upper bound on the optimal cover time of the $\epsilon$-steered random walk. We also demonstrate that our bound on $r$ is best possible up to a constant factor for graphs with strong vertex expansion.

math.CO↗

arXiv2026

Linear equations and chromatic thresholds in $B_h$ sets

We derive sparse analogs of several Roth-type results, showing that they hold in $B_h$ sets of near-maximum size. It is shown that if a $B_h$ set is free of pairwise distinct solutions to a linear equation with more than $2h$ variables then it must be a constant factor smaller than the best-known upper bound on the size of any $B_h$ set. As a key input, it is established that extremal $B_h$ sets are Fourier pseudorandom. If the forbidden equation has a certain subdivision structure, an asymptotic saving is obtained. The case of Sidon sets ($h=2$) was previously studied by Conlon, Fox, Sudakov, and Zhao as well as Prendiville. When forbidding a non-translation-invariant equation $E$ from a Sidon set, it is shown that if $E$ has a zero-sum subcollection of at least five coefficients then the Sidon set must either be very small or generate a Cayley graph with bounded chromatic number. On the other hand, large Sidon sets are constructed that generate Cayley graphs with unbounded chromatic number and are also free of multiple equations with zero-sum subcollections of four coefficients. This can be viewed as a sparse analog of a result of Liu, Wu, Yang, and Zhang characterizing linear equations with vanishing chromatic threshold.

math.CO↗

arXiv2026

Coloring sparse random Cayley graphs

It is shown that there exists $c > 0$ so that the Cayley graph over any finite abelian group $Z$ generated by $c \log |Z|$ random elements is properly 3-colorable with high probability (as $|Z| \to \infty$). This is asymptotically tight and improves the best-known bound due to Alon of $\frac{1}{4}\log \log |Z|$ elements. It also settles the abelian case of Alon's suggestion that a bound of $c \log |G|$ may hold for any finite solvable group $G$.

math.CO↗

arXiv2026

Randomly piercing algebraic sets

We show, for example, that if one samples \[\frac{\log p}{2\log(1+(p-1)^{-1})} \cdot n^2(1 + o_{n\to \infty}(1))\] points in $\mathbb{F}_p^n$ at random then asymptotically almost surely this set intersects every quadratic hypersurface. We furthermore show that this is tight in that sampling $o_{n\to\infty}(n^2)$ fewer points almost surely fails to intersect some quadratic hypersurface. Our main result is a sharp threshold for the following problem: how many points in $\mathbb{F}_p^n$ does one need to randomly sample to almost surely intersect every algebraic set defined by at most $s$ polynomials each of degree at most $k$? As an application we improve lower bounds in the random Szemer\'{e}di theorem in $\mathbb{F}_p^n$, in particular obtaining a leading constant which grows as the threshold for what is considered a `dense' set in Szemer\'{e}di's theorem shrinks.

math.NT↗

arXiv2025

New Sidorenko-type inequalities in tournaments

As a directed analog of Sidorenko's conjecture in extremal graph theory, Fox, Himwich, Zhou, and the second author defined an oriented graph $H$ to be tournament Sidorenko (anti-Sidorenko) if the random tournament asymptotically minimizes (maximizes) the number of copies of $H$ among all tournaments. We prove new inequalities of this form for oriented trees and cycles, considering both local and global notions of the Sidorenko property. We make progress on a conjecture of the aforementioned authors that every tree has an anti-Sidorenko direction, and give a characterization of short paths. For long paths we show that orientations are split symmetrically between being locally Sidorenko and anti-Sidorenko, yet almost all orientations are not globally Sidorenko. Finally, we give algorithms characterizing the local Sidorenko status of paths and cycles when the number of vertices is not divisible by four.

math.CO↗

arXiv2025

Cutting a unit square and permuting blocks

Consider a random permutation of $kn$ objects that permutes $n$ disjoint blocks of size $k$ and then permutes elements within each block. Normalizing its cycle lengths by $kn$ gives a random partition of unity, and we derive the limit law of this partition as $k,n \to \infty$. The limit may be constructed via a simple square cutting procedure that generalizes stick breaking in the classical case of random permutations ($k=1$). The expected size of the largest part of this square cutting distribution is approximated to be $0.40$, in contrast with the Golomb-Dickman constant around $0.624$ describing the longest cycle of a uniform random permutation as well as the largest prime factor of a random integer. The distribution function of this largest part is shown to also be the mean of a certain multiplicative function. Along the way we give the first extension of the Erd\H{o}s-Tur\'an law to a proper permutation subgroup.

math.CO↗

arXiv2024

Poisson approximation for large permutation groups

Let $G_{k,n}$ be a group of permutations of $kn$ objects which permutes things independently in disjoint blocks of size $k$ and then permutes the blocks. We investigate the probabilistic and/or enumerative aspects of random elements of $G_{k,n}$. This includes novel limit theorems for fixed points, cycles of various lengths, number of cycles and inversions. The limits are compound Poisson distributions with interesting dependence structure.

math.PR↗

arXiv2023

Brain Functional Connectivity Estimation Utilizing Diffusion Kernels on a Structural Connectivity Graph

Functional connectivity (FC) refers to the investigation of interactions between brain regions to understand integration of neural activity in several regions. FC is often estimated using functional magnetic resonance images (fMRI). There has been increasing interest in the potential of multi-modal imaging to obtain robust estimates of FC in high-dimensional settings. We develop novel algorithms adapting graphical methods incorporating diffusion tensor imaging (DTI) to estimate FC with computational efficiency and scalability. We propose leveraging a graphical random walk on DTI to define a new measure of structural connectivity highlighting spurious connected components. Our proposed approach is based on finding appropriate subnetwork topology using permutation testing before selection of subnetwork components comprising FC. Extensive simulations demonstrate that the performance of our methods is comparable to or better than currently used approaches in estimation accuracy, with the advantage of greater speed and simpler implementation. We analyze task-based fMRI data obtained from the Human Connectome Project database using our proposed methods and reveal novel insights into brain interactions during performance of a motor task. We expect that the transparency and flexibility of our approach will prove valuable as further understanding of the structure-function relationship informs future network estimation.

stat.AP↗