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Oanh Nguyen

Publications and source records attributed to Oanh Nguyen.

26 records · Page 2Linked to original sources
arXiv2019

Critical value asymptotics for the contact process on random graphs

Recent progress in the study of the contact process [2] has verified that the extinction-survival threshold $λ_1$ on a Galton-Watson tree is strictly positive if and only if the offspring distribution $ξ$ has an exponential tail. In this paper, we derive the first-order asymptotics of $λ_1$ for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if $ξ$ is appropriately concentrated around its mean, we demonstrate that $λ_1(ξ) \sim 1/\mathbb{E} ξ$ as $\mathbb{E}ξ\rightarrow \infty$, which matches with the known asymptotics on the $d$-regular trees. The same result for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well.

math.PR↗
arXiv2018

On the mixing time of the Diaconis--Gangolli random walk on contingency tables over $\mathbb{Z}/ q \mathbb{Z}$

The Diaconis--Gangolli random walk is an algorithm that generates an almost uniform random graph with prescribed degrees. In this paper, we study the mixing time of the Diaconis--Gangolli random walk restricted on $n\times n$ contingency tables over $\mathbb{Z}/q\mathbb{Z}$. We prove that the random walk exhibits cutoff at $\frac{n^2}{4(1- \cos{\frac{2 π}{q}})} \log n, $ when $\log q=o\left (\frac{\sqrt{\log n}}{\log \log n}\right )$.

math.PR↗
arXiv2017

Roots of random polynomials with coefficients having polynomial growth

In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coeffcients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coeffcients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.

math.PR↗
arXiv2017

Embedding large graphs into a random graph

In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $Δ\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree $Δ$. We show that a random graph $G_{n,p}$ with high probability contains a copy of $H$, provided that $p\gg (n^{-1}\log^{1/Δ}n)^{2/(Δ+1)}$. Our assumption on $p$ is optimal up to the $polylog$ factor. We note that this $polylog$ term matches the conjectured threshold for the spanning case.

math.CO↗
arXiv2016

Packing Loose Hamilton Cycles

A subset $C$ of edges in a $k$-uniform hypergraph $H$ is a \emph{loose Hamilton cycle} if $C$ covers all the vertices of $H$ and there exists a cyclic ordering of these vertices such that the edges in $C$ are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random $k$-uniform hypergraph $H^k_{n,p}$ has vertex set $[n]$ and an edge set $E$ obtained by adding each $k$-tuple $e\in \binom{[n]}{k}$ to $E$ with probability $p$, independently at random. Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but $o(|E|)$ edges, referred to as the \emph{packing problem}. While it is known that the threshold probability for the appearance of a loose Hamilton cycle in $H^k_{n,p}$ is $p=Θ\left(\frac{\log n}{n^{k-1}}\right)$, the best known bounds for the packing problem are around $p=\text{polylog}(n)/n$. Here we make substantial progress and prove the following asymptotically (up to a polylog$(n)$ factor) best possible result: For $p\geq \log^{C}n/n^{k-1}$, a random $k$-uniform hypergraph $H^k_{n,p}$ with high probability contains $N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$ edge-disjoint loose Hamilton cycles. Our proof utilizes and modifies the idea of "online sprinkling" recently introduced by Vu and the first author.

math.CO↗
arXiv2015

Anti-concentration for polynomials of independent random variables

We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. We discuss applications in two different areas. In complexity theory, we prove near optimal lower bounds for computing the Parity, addressing a challenge in complexity theory posed by Razborov and Viola, and also address a problem concerning OR functions. In random graph theory, we derive a general anti-concentration result on the number of copies of a fixed graph in a random graph.

math.PR↗
arXiv2014

On the number of real roots of random polynomials

Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the expectation of the number of real roots is $\frac{2}π \log n + o(\log n)$. In this paper, we determine the true nature of the error term by showing that the expectation equals $\frac{2}π\log n + O(1)$. Prior to this paper, such estimate has been known only in the gaussian case, thanks to works of Edelman and Kostlan.

math.PR↗