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Ashwin Sah

Publications and source records attributed to Ashwin Sah.

At least 19 recordsLinked to original sources
arXiv2026

Majority dynamics on sparse random graphs

Consider a simple undirected graph $G$ on $N$ vertices, with each vertex holding an opinion from one of two options. Starting from the initial opinions on day 1, we run majority dynamics: on each subsequent day, all vertices simultaneously adopt the majority opinion among their neighbors, retaining their current opinion in case of a tie. A well-studied conjecture of Benjamini, Chan, O'Donnell, Tamuz, and Tan states that if $G$ is drawn from the random binomial model $\mathbb{G}(N,p)$, and if the starting opinions are chosen uniformly at random, then as long as $pN \to \infty$ the network will converge to 99% consensus with high probability. We confirm this conjecture as long as $pN \ge N^{\varepsilon}$ for any fixed $\varepsilon > 0$, showing that full unanimity is reached with high probability in $O(1/\varepsilon)$ days. This result improves on a line of results by Fountoulakis, Kang, and Makai, who achieve $\varepsilon = 1/2$; Chakraborti, Kim, Lee, and Tran, who achieve $\varepsilon = 2/5$; and Jaffe, who achieves $\varepsilon = 1/3$. The proof technique involves iteratively revealing the "opinion histories" for each vertex through time, keeping track of the degrees between every vertex and each of $2^k$ opinion history classes on day $k$. Our analysis of this process requires intricate estimates for degree-constrained random graph models using graph enumeration tools from the work of McKay and Wormald as well as Canfield, Greenhill, and McKay, and their extensions by Liebenau and Wormald. In doing so, we connect the discrete dynamics to a deterministic idealized process, whose leading-order behavior is described by conditional Gaussian probabilities and expectations.

math.PR↗
arXiv2026

A tight bound for affine-linearity, via universal ballot matrices

Based on work with Greenfeld and with Ziegler, Tao showed a concatenation result that if a map $f : \mathbb{F}^2 \to \mathbb{F}$ is affine-linear on every line parallel to the coordinate axes, and on all lines with a fixed nonzero slope (where the field $\mathbb{F}$ has size $> 2$), then $f$ is affine-linear on $\mathbb{F}^2$. We extend this from $\mathbb{F}^2$ to $\mathbb{F}^n$ and obtain a tight minimum number of additional lines needed -- $N = \binom{n}{\lfloor n/2 \rfloor}$ -- for every field $\mathbb{F}$ with $3 \leqslant n < |\mathbb{F}|$. The proof is constructive and shows a stronger result: the existence of a universal family of $0$-$1$ matrices of size $\binom{n}{k} \times \binom{n}{k}$ (one for each pair $0 \leqslant k \leqslant n$), which are indexed by ballot sets and are unimodular over all unital commutative rings. We also show a second tightness: of the assumption $n < |\mathbb{F}|$. Else, there exist multi-affine maps $f$ which are affine-linear on every line through the origin, but not affine-linear globally on $\mathbb{F}^n$. More strongly, we prove this dichotomy -- including the bound of $N$ -- over all integral domains, or Noetherian (e.g.\ finite or Artinian) rings, or products of these. This yields a novel numerical invariant for affine-linearity, for every product of Noetherian rings and integral domains.

math.CO↗
arXiv2025

The limiting spectral law for sparse iid matrices

Let $A$ be an $n\times n$ matrix with iid entries where $A_{ij} \sim \mathrm{Ber}(p)$ is a Bernoulli random variable with parameter $p = d/n$. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution as $n \rightarrow \infty$. This essentially resolves a long line of work to determine the spectral laws of iid matrices and is the first known example for non-Hermitian random matrices at this level of sparsity.

math.PR↗
arXiv2025

A central limit theorem for the matching number of a sparse random graph

In 1981, Karp and Sipser proved a law of large numbers for the matching number of a sparse Erdős-Rényi random graph, in an influential paper pioneering the so-called differential equation method for analysis of random graph processes. Strengthening this classical result, and answering a question of Aronson, Frieze and Pittel, we prove a central limit theorem in the same setting: the fluctuations in the matching number of a sparse random graph are asymptotically Gaussian. Our new contribution is to prove this central limit theorem in the subcritical and critical regimes, according to a celebrated algorithmic phase transition first observed by Karp and Sipser. Indeed, in the supercritical regime, a central limit theorem has recently been proved in the PhD thesis of Kreačić, using a stochastic generalisation of the differential equation method (comparing the so-called Karp-Sipser process to a system of stochastic differential equations). Our proof builds on these methods, and introduces new techniques to handle certain degeneracies present in the subcritical and critical cases. Curiously, our new techniques lead to a non-constructive result: we are able to characterise the fluctuations of the matching number around its mean, despite these fluctuations being much smaller than the error terms in our best estimates of the mean. We also prove a central limit theorem for the rank of the adjacency matrix of a sparse random graph.

math.CO↗
arXiv2025

The sparse circular law, revisited

Let $A_n$ be an $n\times n$ matrix with iid entries distributed as Bernoulli random variables with parameter $p = p_n$. Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of $A_n \cdot (pn)^{-1/2}$ is approximately uniform on the unit disk as $n\rightarrow \infty$ as long as $pn \rightarrow \infty$, which is the natural necessary condition. In this paper we give a much simpler proof of this result, in its full generality, using a perspective we developed in our recent proof of the existence of the limiting spectral law when $pn$ is bounded. One feature of our proof is that it avoids the use of $ε$-nets entirely and, instead, proceeds by studying the evolution of the singular values of the shifted matrices $A_n-zI$ as we incrementally expose the randomness in the matrix.

math.PR↗
arXiv2025

On the Spielman-Teng Conjecture

Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $σ_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( σ_{n}(M) \leq \varepsilon n^{-1/2} \big) = (1+o(1)) \varepsilon + e^{-Ω(n)}\] for all $0 \leq \varepsilon \ll 1$. This resolves, up to a $1+o(1)$ factor, a seminal conjecture of Spielman and Teng.

math.PR↗
arXiv2024

Local limit theorem for joint subgraph counts

Extending a previous result of the first two authors, we prove a local limit theorem for the joint distribution of subgraph counts in the Erdős-Rényi random graph $G(n,p)$. This limit can be described as a nonlinear transformation of a multivariate normal distribution, where the components of the multivariate normal correspond to the graph factors of Janson. As an application, we show a number of results concerning the existence and enumeration of proportional graphs and related concepts, answering various questions of Janson and collaborators in the affirmative.

math.PR↗
arXiv2024

Quasipolynomial bounds on the inverse theorem for the Gowers $U^{s+1}[N]$-norm

We prove quasipolynomial bounds on the inverse theorem for the Gowers $U^{s+1}[N]$-norm. The proof is modeled after work of Green, Tao, and Ziegler and uses as a crucial input recent work of the first author regarding the equidistribution of nilsequences. In a companion paper, this result will be used to improve the bounds on Szemerédi's theorem.

math.CO↗
arXiv2024

Anticoncentration in Ramsey graphs and a proof of the Erdős-McKay conjecture

An $n$-vertex graph is called $C$-Ramsey if it has no clique or independent set of size $C\log_2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge-statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a $C$-Ramsey graph. This brings together two ongoing lines of research: the study of "random-like" properties of Ramsey graphs and the study of small-ball probabilities for low-degree polynomials of independent random variables. The proof proceeds via an "additive structure" dichotomy on the degree sequence, and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics, and low-rank approximation. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős offered one of his notorious monetary prizes.

math.CO↗
arXiv2024

Enumerating Matroids and Linear Spaces

We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the puzzle for enumerating fixed-rank matroids at this level of accuracy: the numbers of rank-1 and rank-2 matroids on a ground set of size $n$ have exact representations in terms of well-known combinatorial functions, and it was recently proved by van der Hofstad, Pendavingh, and van der Pol that for constant $r\ge 4$ there are $(e^{1-r}n+o(n))^{n^{r-1}/r!}$ rank-$r$ matroids on a ground set of size $n$. In our proof, we introduce a new approach for bounding the number of clique decompositions of a complete graph, using quasirandomness instead of the so-called entropy method that is common in this area.

math.CO↗
arXiv2024

Improved bounds for five-term arithmetic progressions

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp((\log\log N)^{c})}.\] Our work is a consequence of recent improved bounds on the $U^4$-inverse theorem of the first author and the fact that $3$-step nilsequences may be approximated by locally cubic functions on shifted Bohr sets. This combined with the density increment strategy of Heath-Brown and Szemer{é}di, codified by Green and Tao, gives the desired result.

math.NT↗
arXiv2024

Singularity of the k-core of a random graph

Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $k\ge 3$ and $λ> 0$, an Erdős--Rényi random graph $G\sim\mathbb{G}(n,λ/n)$ with $n$ vertices and edge probability $λ/n$ typically has the property that its $k$-core (its largest subgraph with minimum degree at least $k$) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for "extremely sparse'' random matrices with density $O(1/n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to "boost'' the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.

math.PR↗
arXiv2024

Improved Bounds for Szemerédi's Theorem

Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that \[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\] Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemerédi as reformulated by Green and Tao.

math.CO↗
arXiv2023

A Toolkit for Robust Thresholds

Consider a host hypergraph $G$ which contains a spanning structure due to minimum degree considerations. We collect three results proving that if the edges of $G$ are sampled at the appropriate rate then the spanning structure still appears with high probability in the sampled hypergraph. We prove such results for perfect matchings in hypergraphs above Dirac thresholds, for $K_r$-factors in graphs satisfying the Hajnal--Szemerédi minimum degree condition, and for bounded-degree spanning trees. In each case our proof is based on constructing a spread measure and then applying recent results on the (fractional) Kahn--Kalai conjecture connecting the existence of such measures with probabilistic thresholds. For our second result we give a shorter and more general proof of a recent theorem of Allen, Böttcher, Corsten, Davies, Jenssen, Morris, Roberts, and Skokan which handles the $r=3$ case with different techniques. In particular, we answer a question of theirs with regards to the number of $K_r$-factors in graphs satisfying the Hajnal--Szemerédi minimum degree condition.

math.CO↗
arXiv2023

Subgraph distributions in dense random regular graphs

Given connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of McKay from the 2010 International Congress of Mathematicians. In fact, we prove that the behavior of the variance of the number of copies of $H$ depends in a delicate manner on the occurrence and number of cycles of length $3,4,5$ as well as paths of length $3$ in $H$. More generally, we provide control of the asymptotic distribution of certain statistics of bounded degree which are invariant under vertex permutations, including moments of the spectrum of a random regular graph. Our techniques are based on combining complex-analytic methods due to McKay and Wormald used to enumerate regular graphs with the notion of graph factors developed by Janson in the context of studying subgraph counts in $\mathbb{G}(n,p)$.

math.CO↗
arXiv2023

On Perfectly Friendly Bisections of Random Graphs

We prove that there exists a constant $γ_{\mathrm{crit}}\approx .17566$ such that if $G\sim \mathbb{G}(n,1/2)$ then for any $\varepsilon > 0$ with high probability $G$ has a equipartition such that each vertex has $(γ_{\mathrm{crit}}-\varepsilon)\sqrt{n}$ more neighbors in its own part than in the other part and with high probability no such partition exists for a separation of $(γ_{\mathrm{crit}}+\varepsilon)\sqrt{n}$. The proof involves a number of tools ranging from isoperimetric results on vertex-transitive sets of graphs coming from Boolean functions, switchings, degree enumeration formulas, and the second moment method. Our results substantially strengthen recent work of Ferber, Kwan, Narayanan, and the last two authors on a conjecture of Füredi from 1988 and in particular prove the existence of fully-friendly bisections in $\mathbb{G}(n,1/2)$

math.PR↗