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Ashwin Sah

Publications and source records attributed to Ashwin Sah.

At least 37 records · Page 2Linked to original sources
arXiv2023

The Exact Rank of Sparse Random Graphs

Two landmark results in combinatorial random matrix theory, due to Komlós and Costello-Tao-Vu, show that discrete random matrices and symmetric discrete random matrices are typically nonsingular. In particular, in the language of graph theory, when $p$ is a fixed constant, the biadjacency matrix of a random Erdős-Rényi bipartite graph $\mathbb{G}(n,n,p)$ and the adjacency matrix of an Erdős-Rényi random graph $\mathbb{G}(n,p)$ are both nonsingular with high probability. However, very sparse random graphs (i.e., where $p$ is allowed to decay rapidly with $n$) are typically singular, due to the presence of "local" dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbour. In this paper we give a combinatorial description of the rank of a sparse random graph $\mathbb{G}(n,n,c/n)$ or $\mathbb{G}(n,c/n)$ in terms of such local dependencies, for all constants $c\ne e$ (and we present some evidence that the situation is very different for $c=e$). This gives an essentially complete answer to a question raised by Vu at the 2014 International Congress of Mathematicians. As applications of our main theorem and its proof, we also determine the asymptotic singularity probability of the 2-core of a sparse random graph, we show that the rank of a sparse random graph is extremely well-approximated by its matching number, and we deduce a central limit theorem for the rank of $\mathbb{G}(n,c/n)$.

math.CO↗
arXiv2023

The intransitive dice kernel: $\frac{\mathbf{1}_{x\ge y}-\mathbf{1}_{x\le y}}{4} - \frac{3(x-y)(1+xy)}{8}$

Answering a pair of questions of Conrey, Gabbard, Grant, Liu, and Morrison, we prove that a triplet of dice drawn from the multiset model are intransitive with probability $1/4+o(1)$ and the probability a random pair of dice tie tends toward $αn^{-1}$ for an explicitly defined constant $α$. This extends and sharpens the recent results of Polymath regarding the balanced sequence model. We further show the distribution of larger tournaments converges to a universal tournamenton in both models. This limit naturally arises from the discrete spectrum of a certain skew-symmetric operator (given by the kernel in the title acting on $L^2([-1,1])$). The limit exhibits a degree of symmetry and can be used to prove that, for instance, the limiting probability that $A_i$ beats $A_{i+1}$ for $1\le i\le 4$ and that $A_5$ beats $A_1$ is $1/32+o(1)$. Furthermore, the limiting tournamenton has range contained in the discrete set $\{0,1\}$. This proves that the associated tournamenton is non-quasirandom in a dramatic fashion, vastly extending work of Cornacchia and Hązła regarding the continuous analogue of the balanced sequence model. The proof is based on a reduction to conditional central limit theorems (related to work of Polymath), the use of a "Poissonization" style method to reduce to computations with independent random variables, and the systematic use of switching-based arguments to extract cancellation in Fourier estimates when establishing local limit-type estimates.

math.PR↗
arXiv2023

The existence of subspace designs

We prove the existence of subspace designs with any given parameters, provided that the dimension of the underlying space is sufficiently large in terms of the other parameters of the design and satisfies the obvious necessary divisibility conditions. This settles an open problem from the 1970s. Moreover, we also obtain an approximate formula for the number of such designs.

math.CO↗
arXiv2022

Sparse recovery properties of discrete random matrices

Motivated by problems from compressed sensing, we determine the threshold behavior of a random $n\times d$ $\pm 1$ matrix $M_{n,d}$ with respect to the property "every $s$ columns are linearly independent". In particular, we show that for every $0<δ<1$ and $s=(1-δ)n$, if $d\leq n^{1+1/2(1-δ)-o(1)}$ then with high probability every $s$ columns of $M_{n,d}$ are linearly independent, and if $d\geq n^{1+1/2(1-δ)+o(1)}$ then with high probability there are some $s$ linearly dependent columns.

math.CO↗
arXiv2022

Substructures in Latin squares

We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order-$n$ Latin squares with no intercalate (i.e., no $2\times2$ Latin subsquare) is at least $(e^{-9/4}n-o(n))^{n^{2}}$. Equivalently, $\mathbb{P}\left[\mathbf{N}=0\right]\ge e^{-n^{2}/4-o(n^{2})}=e^{-(1+o(1))\mathbb{E}\mathbf{N}}$, where $\mathbf{N}$ is the number of intercalates in a uniformly random order-$n$ Latin square. In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general large-deviation problem for intercalates in random Latin squares, up to constant factors in the exponent: for any constant $0<δ\le1$ we have $\mathbb{P}[\mathbf{N}\le(1-δ)\mathbb{E}\mathbf{N}]=\exp(-Θ(n^{2}))$ and for any constant $δ>0$ we have $\mathbb{P}[\mathbf{N}\ge(1+δ)\mathbb{E}\mathbf{N}]=\exp(-Θ(n^{4/3}\log n))$. Finally, as an application of some new general tools for studying substructures in random Latin squares, we show that in almost all order-$n$ Latin squares, the number of cuboctahedra (i.e., the number of pairs of possibly degenerate $2\times2$ submatrices with the same arrangement of symbols) is of order $n^{4}$, which is the minimum possible. As observed by Gowers and Long, this number can be interpreted as measuring ``how associative'' the quasigroup associated with the Latin square is.

math.CO↗
arXiv2022

Cayley graphs that have a quantum ergodic eigenbasis

We investigate which finite Cayley graphs admit a quantum ergodic eigenbasis, proving that this holds for any Cayley graph on a group of size $n$ for which the sum of the dimensions of its irreducible representations is $o(n)$, yet there exist Cayley graphs that do not have any quantum ergodic eigenbasis.

math.SP↗
arXiv2022

Spencer's theorem in nearly input-sparsity time

A celebrated theorem of Spencer states that for every set system $S_1,\dots, S_m \subseteq [n]$, there is a coloring of the ground set with $\{\pm 1\}$ with discrepancy $O(\sqrt{n\log(m/n+2)})$. We provide an algorithm to find such a coloring in near input-sparsity time $\tilde{O}(n+\sum_{i=1}^{m}|S_i|)$. A key ingredient in our work, which may be of independent interest, is a novel width reduction technique for solving linear programs, not of covering/packing type, in near input-sparsity time using the multiplicative weights update method.

cs.DS↗
arXiv2022

Threshold for Steiner triple systems

We prove that with high probability $\mathbb{G}^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system for $n\equiv 1,3\pmod{6}$, establishing the exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.

math.CO↗
arXiv2022

Optimal minimization of the covariance loss

Let $X$ be a random vector valued in $\mathbb{R}^{m}$ such that $\|X\|_{2} \le 1$ almost surely. For every $k\ge 3$, we show that there exists a sigma algebra $\mathcal{F}$ generated by a partition of $\mathbb{R}^{m}$ into $k$ sets such that \[\|\operatorname{Cov}(X) - \operatorname{Cov}(\mathbb{E}[X\mid\mathcal{F}]) \|_{\mathrm{F}} \lesssim \frac{1}{\sqrt{\log{k}}}.\] This is optimal up to the implicit constant and improves on a previous bound due to Boedihardjo, Strohmer, and Vershynin. Our proof provides an efficient algorithm for constructing $\mathcal{F}$ and leads to improved accuracy guarantees for $k$-anonymous or differentially private synthetic data. We also establish a connection between the above problem of minimizing the covariance loss and the pinning lemma from statistical physics, providing an alternate (and much simpler) algorithmic proof in the important case when $X \in \{\pm 1\}^m/\sqrt{m}$ almost surely.

math.PR↗
arXiv2022

Enumerating coprime permutations

Define a permutation $σ$ to be coprime if $\gcd(m,σ(m)) = 1$ for $m\in[n]$. In this note, proving a recent conjecture of Pomerance, we prove that the number of coprime permutations on $[n]$ is $n!\cdot (c+o(1))^n$ where \[c = \prod_{p\text{ prime }}\frac{(p-1)^{2(1-1/p)}}{p\cdot (p-2)^{(1-2/p)}}.\] The techniques involve entropy maximization for the upper bound, and a mixture of number-theoretic bounds, permanent estimates, and the absorbing method for the lower bound.

math.NT↗
arXiv2021

Large deviations in random Latin squares

In this note, we study large deviations of the number $\mathbf{N}$ of intercalates ($2\times2$ combinatorial subsquares which are themselves Latin squares) in a random $n\times n$ Latin square. In particular, for constant $δ>0$ we prove that $\Pr(\mathbf{N}\le(1-δ)n^{2}/4)\le\exp(-Ω(n^{2}))$ and $\Pr(\mathbf{N}\ge(1+δ)n^{2}/4)\le\exp(-Ω(n^{4/3}(\log n)^{2/3}))$, both of which are sharp up to logarithmic factors in their exponents. As a consequence, we deduce that a typical order-$n$ Latin square has $(1+o(1))n^{2}/4$ intercalates, matching a lower bound due to Kwan and Sudakov and resolving an old conjecture of McKay and Wanless.

math.CO↗
arXiv2021

A counterexample to the Bollobás-Riordan conjectures on sparse graph limits

Bollobás and Riordan, in their paper "Metrics for sparse graphs," proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom $C_4$-counts), but with no limit expressible as a kernel.

math.CO↗
arXiv2021

Online Edge Coloring via Tree Recurrences and Correlation Decay

We give an online algorithm that with high probability computes a $\left(\frac{e}{e-1} + o(1)\right)Δ$ edge coloring on a graph $G$ with maximum degree $Δ= ω(\log n)$ under online edge arrivals against oblivious adversaries, making first progress on the conjecture of Bar-Noy, Motwani, and Naor in this general setting. Our algorithm is based on reducing to a matching problem on locally treelike graphs, and then applying a tree recurrences based approach for arguing correlation decay.

cs.DS↗
arXiv2021

Note on random Latin squares and the triangle removal process

This is a companion note to the paper "Almost all Steiner triple systems have perfect matchings (arXiv:1611.02246). That paper contains several general lemmas about random Steiner triple systems; in this note we record analogues of these lemmas for random Latin squares, which in particular are necessary ingredients for our recent paper "Large deviations in random Latin squares" (arXiv:2106.11932). Most important is a relationship between uniformly random order-$n$ Latin squares and the triangle removal process on the complete tripartite graph $K_{n,n,n}$.

math.CO↗
arXiv2021

Approximate counting and sampling via local central limit theorems

We give an FPTAS for computing the number of matchings of size $k$ in a graph $G$ of maximum degree $Δ$ on $n$ vertices, for all $k \le (1-δ)m^*(G)$, where $δ>0$ is fixed and $m^*(G)$ is the matching number of $G$, and an FPTAS for the number of independent sets of size $k \le (1-δ) α_c(Δ) n$, where $α_c(Δ)$ is the NP-hardness threshold for this problem. We also provide quasi-linear time randomized algorithms to approximately sample from the uniform distribution on matchings of size $k \leq (1-δ)m^*(G)$ and independent sets of size $k \leq (1-δ)α_c(Δ)n$. Our results are based on a new framework for exploiting local central limit theorems as an algorithmic tool. We use a combination of Fourier inversion, probabilistic estimates, and the deterministic approximation of partition functions at complex activities to extract approximations of the coefficients of the partition function. For our results for independent sets, we prove a new local central limit theorem for the hard-core model that applies to all fugacities below $λ_c(Δ)$, the uniqueness threshold on the infinite $Δ$-regular tree.

cs.DS↗
arXiv2021

Patterns without a popular difference

Which finite sets $P \subseteq \mathbb{Z}^r$ with $|P| \ge 3$ have the following property: for every $A \subseteq [N]^r$, there is some nonzero integer $d$ such that $A$ contains $(α^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p : p \in P\}$, where $α= |A|/N^r$? Green showed that all 3-point $P \subseteq \mathbb{Z}$ have the above property. Green and Tao showed that 4-point sets of the form $P = \{a, a+b, a+c, a+b+c\} \subseteq \mathbb{Z}$ also have the property. We show that no other sets have the above property. Furthermore, for various $P$, we provide new upper bounds on the number of translates of $d \cdot P$ that one can guarantee to find.

math.CO↗