Searcharxiv⌕ Search

arXiv subjects

Pu Gao

Publications and source records attributed to Pu Gao.

At least 19 recordsLinked to original sources
arXiv2026

Non-uniform Kahn-Kalai, spread, variants, and applications

Building on B.Park and Vondrak's recent generalization of the J.Park-Pham Theorem (formerly known as Kahn-Kalai conjecture) to non-uniform probability measures, this paper introduces the notion of "spread" for the non-uniform setting. This provides a framework to establish 1-statements for subgraph containment in inhomogeneous random graphs with or without a set of forced edges. Using this approach, we derived conditions for the emergence of perfect matchings in the Stochastic Block Model and the Chung-Lu model, and verified that these conditions are in general not tight, but they capture thresholds across a broad range of regimes. Finally, we bridge this non-uniform framework with $\mathcal{G}(n,\textbf{d})$, utilizing a coupling argument to demonstrate thresholds for perfect matchings in $\mathcal{G}(n,\textbf{d})$ for a broad range of degree sequences $\textbf{d}$.

math.CO↗
arXiv2025

The satisfiability threshold and solution space of random uniquely extendable constraint satisfaction problems

We study the satisfiability threshold and solution-space geometry of random constraint satisfaction problems defined over uniquely extendable (UE) constraints. Motivated by a conjecture of Connamacher and Molloy, we consider random $k$-ary UE-SAT instances in which each constraint function is drawn, according to a certain distribution $\pi$, from a specified subset of uniquely extendable constraints over an $r$-spin set. We introduce a flexible model $H_n(\pi,k,m)$ that allows arbitrary distributions $\pi$ on constraint types, encompassing both random linear systems and previously studied UE-SAT models. Our main result determines the satisfiability threshold for a wide family of distributions $\pi$. Under natural reducibility or symmetry conditions on $\operatorname{supp}(\pi)$, we prove that the satisfiability threshold of $H_n(\pi,k,m)$ coincides with the classical $k$-XORSAT threshold.

math.CO↗
arXiv2025

Sandwiching between random regular graphs and Erd\H{o}s-R\'enyi graphs: configuration model and unions of perfect matchings

We establish new couplings among several random graph and multigraph models related to the random regular graph $G(n,d)$, including the configuration model and unions of random perfect matchings. As a main result, we verify the Kim-Vusandwich conjecture for all large degrees $d=n-O(\log^4 n)$ and prove a weakened version for $d=O(\log^4 n)$, which are the only remaining open cases. Our approach introduces a coupling framework that links $G(n,d)$ and $G(n,p)$ through a chain of intermediate models.

math.CO↗
arXiv2025

Towards Emotionally Consistent Text-Based Speech Editing: Introducing EmoCorrector and The ECD-TSE Dataset

Text-based speech editing (TSE) modifies speech using only text, eliminating re-recording. However, existing TSE methods, mainly focus on the content accuracy and acoustic consistency of synthetic speech segments, and often overlook the emotional shifts or inconsistency issues introduced by text changes. To address this issue, we propose EmoCorrector, a novel post-correction scheme for TSE. EmoCorrector leverages Retrieval-Augmented Generation (RAG) by extracting the edited text's emotional features, retrieving speech samples with matching emotions, and synthesizing speech that aligns with the desired emotion while preserving the speaker's identity and quality. To support the training and evaluation of emotional consistency modeling in TSE, we pioneer the benchmarking Emotion Correction Dataset for TSE (ECD-TSE). The prominent aspect of ECD-TSE is its inclusion of $<$text, speech$>$ paired data featuring diverse text variations and a range of emotional expressions. Subjective and objective experiments and comprehensive analysis on ECD-TSE confirm that EmoCorrector significantly enhances the expression of intended emotion while addressing emotion inconsistency limitations in current TSE methods. Code and audio examples are available at https://github.com/AI-S2-Lab/EmoCorrector.

eess.AS↗
arXiv2025

The dimension of sparse and co-sparse random graph orders

A random graph order is a partial order obtained from a random graph on $[n]$ by taking the transitive closure of the adjacency relation. The dimension of the random graph orders from random bipartite graphs $B(n,n,p)$ and from $G(n,p)$ were previously studied when $p=\Omega(\log n/n)$ and when $p$ is not too close to 1. There is a conjectured phase transition in the sparse range at $p=1/n$. In this paper, we investigate this conjectured phase transition and estimate the dimension of the partial orders arising from $B(n,n,p)$ and $G(n,p)$ when $p=O(1/n)$. For the random bipartite order, we additionally estimate its dimension in the co-sparse regime, thereby closing all previously open ranges of $p$. Finally, we establish a general upper bound on the dimension of partial orders based on their decompositions into suborders, a result that is of independent interest.

math.CO↗
arXiv2024

Evolution of random representable matroids: minors, circuits, connectivity and the critical number

We study the evolution of random matroids represented by the sequence of random matrices over ${\mathbb F}_q$ where columns are added one after the other, and each column vector is a uniformly random vector in ${\mathbb F}_q^n$, independent of each other. We study the appearance of matroid minors, the appearance of circuits, the evolution of the connectivities and the critical number. We settle several open problems in the literature.

math.CO↗
arXiv2024

Minors of matroids represented by sparse random matrices over finite fields

Consider a random $n\times m$ matrix $A$ over the finite field of order $q$ where every column has precisely $k$ nonzero elements, and let $M[A]$ be the matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden (RS\&A 2019) proved that given a fixed binary matroid $N$, if $k\ge k_N$ and $m/n\ge d_N$ where $k_N$ and $d_N$ are sufficiently large constants depending on N, then a.a.s. $M[A]$ contains $N$ as a minor. We improve their result by determining the sharp threshold (of $m/n$) for the appearance of a fixed matroid $N$ as a minor of $M[A]$, for every $k\ge 3$, and every finite field.

math.CO↗
arXiv2023

Building Hamiltonian Cycles in the Semi-Random Graph Process in Less Than $2n$ Rounds

The semi-random graph process is an adaptive random graph process in which an online algorithm is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible. We focus on the property of Hamiltonicity. We present an adaptive strategy which creates a Hamiltonian cycle in $\alpha n$ rounds, where $\alpha < 1.81696$ is derived from the solution to a system of differential equations. We also show that achieving Hamiltonicity requires at least $\beta n$ rounds, where $\beta > 1.26575$.

math.CO↗
arXiv2023

On the pre- and post-positional semi-random graph processes

We study the semi-random graph process, and a variant process recently suggested by Nick Wormald. We show that these two processes are asymptotically equally fast in constructing a semi-random graph $G$ that has property ${\mathcal P}$, for the following examples of ${\mathcal P}$: - ${\mathcal P}$ is the set of graphs containing a $d$-degenerate subgraph, where $d\ge 1$ is fixed; - ${\mathcal P}$ is the set of $k$-connected graphs, where $k\ge 1$ is fixed. In particular, our result of the $k$-connectedness above settles the open case $k=2$ of the original semi-random graph process. We also prove that there exist properties ${\mathcal P}$ where the two semi-random graph processes do not construct a graph in ${\mathcal P}$ asymptotically equally fast. We further propose some conjectures on ${\mathcal P}$ for which the two processes perform differently.

math.CO↗
arXiv2023

Embedding theorems for random graphs with specified degrees

Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let $\mathcal{G}(n,W)$ be the random graph obtained by independently including each edge $jk$ with probability $W_{jk}$. Given a degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $\mathcal{G}(n,{\bf d})$ denote a uniformly random graph with degree sequence ${\bf d}$. We couple $\mathcal{G}(n,W)$ and $\mathcal{G}(n,{\bf d})$ together so that a.a.s. $\mathcal{G}(n,W)$ is a subgraph of $\mathcal{G}(n,{\bf d})$, where $W$ is some function of ${\bf d}$. Let $\Delta({\bf d})$ denote the maximum degree in ${\bf d}$. Our coupling result is optimal when $\Delta({\bf d})^2\ll \|{\bf d}\|_1$, i.e.\ $W_{ij}$ is asymptotic to $\mathbb{P}(ij\in \mathcal{G}(n,{\bf d}))$ for every $i,j\in [n]$. We also have coupling results for ${\bf d}$ that are not constrained by the condition $\Delta({\bf d})^2\ll \|{\bf d}\|_1$. For such ${\bf d}$ our coupling result is still close to optimal, in the sense that $W_{ij}$ is asymptotic to $\mathbb{P}(ij\in \mathcal{G}(n,{\bf d}))$ for most pairs $i,j\in [n]$.

math.CO↗
arXiv2022

A Fully Adaptive Strategy for Hamiltonian Cycles in the Semi-Random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a Hamiltonian cycle in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves it in $αn$ rounds, where $α< 2.01678$ is derived from the solution to some system of differential equations. We also show that the player cannot achieve the desired property in less than $βn$ rounds, where $β> 1.26575$. These results improve the previously best known bounds and, as a result, the gap between the upper and lower bounds is decreased from 1.39162 to 0.75102.

math.CO↗
arXiv2022

Sandwiching random regular graphs between binomial random graphs

Kim and Vu made the following conjecture (\textit{Advances in Mathematics}, 2004): if $d\gg \log n$, then the random $d$-regular graph $\mathcal G(n,d)$ can asymptotically almost surely be "sandwiched" between $\mathcal G(n,p_1)$ and $\mathcal G(n,p_2)$ where $p_1$ and $p_2$ are both $(1+o(1))d/n$. They proved this conjecture for $\log n\ll d\le n^{1/3-o(1)}$, with a defect in the sandwiching: $\mathcal G(n,d)$ contains $\mathcal G(n,p_1)$ perfectly, but is not completely contained in $\mathcal G(n,p_2)$. Recently, the embedding $\mathcal G(n,p_1) \subseteq \mathcal G(n,d)$ was improved by Dudek, Frieze, Ruciński and Šileikis to $d=o(n)$. In this paper, we prove Kim--Vu's sandwich conjecture, with perfect containment on both sides, for all $d\gg n/\sqrt{\log n}$. For $d=O(n/\sqrt{\log n})$, we prove a weaker version of the sandwich conjecture with $p_2$ approximately equal to $(d/n)\log n$, without any defect. In addition to sandwiching regular graphs, our results cover graphs whose degrees are asymptotically equal. The proofs rely on estimates for the probability that a random factor of a pseudorandom graph contains a given edge, which is of independent interest. As applications, we obtain new results on the properties of random graphs with given near-regular degree sequences, including Hamiltonicity and universality in subgraph containment. We also determine several graph parameters in these random graphs, such as the chromatic number, small subgraph counts, the diameter, and the independence number. We are also able to characterise many phase transitions in edge percolation on these random graphs, such as the threshold for the appearance of a giant component.

math.CO↗
arXiv2022

The satisfiability threshold for random linear equations

Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ be a random vector chosen independently of $A$. We identify the threshold $m/n$ up to which the linear system $A x=y$ has a solution with high probability and analyse the geometry of the set of solutions. In the special case $q=2$, known as the random $k$-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof technique was subsequently extended to the cases $q=3,4$ [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to $q>3$. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.

math.CO↗
arXiv2022

The number of perfect matchings, and the nesting properties, of random regular graphs

We prove that the number of perfect matchings in ${\mathcal G}(n,d)$ is asymptotically normal when $n$ is even, $d\to\infty$ as $n\to\infty$, and $d=O(n^{1/7}/\log^2 n)$. This is the first distributional result of spanning subgraphs of ${\mathcal G}(n,d)$ when $d\to\infty$. Moreover, we prove that ${\mathcal G}(n,d-1)$ and ${\mathcal G}(n,d)$ can be coupled so that ${\mathcal G}(n,d-1)$ is a subgraph of ${\mathcal G}(n,d)$ with high probability when $d\to\infty$ and $d=o(n^{1/3})$. Further, if $d=Ω(\log^7 n)$, $d=O(n^{1/7}/\log^2n)$, and $d\le d'\le n-1$ then ${\mathcal G}(n,d)$ and ${\mathcal G}(n,d')$ can be coupled so that asymptotically almost surely ${\mathcal G}(n,d)$ is a subgraph of ${\mathcal G}(n,d')$.

math.CO↗
arXiv2022

Perfect Matchings in the Semi-random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a perfect matching in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves a perfect matching in $βn$ rounds, where the value of $β< 1.206$ is derived from a solution to some system of differential equations. This improves upon the previously best known upper bound of $(1+2/e+o(1)) \, n < 1.736 \, n$ rounds. We also improve the previously best lower bound of $(\ln 2 + o(1)) \, n > 0.693 \, n$ and show that the player cannot achieve the desired property in less than $αn$ rounds, where the value of $α> 0.932$ is derived from a solution to another system of differential equations. As a result, the gap between the upper and lower bounds is decreased roughly four times.

math.CO↗
arXiv2022

The full rank condition for sparse random matrices

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.

math.CO↗
arXiv2021

Linear-time uniform generation of random sparse contingency tables with specified marginals

We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics. When $Δ^4< M/5$, where $Δ$ is the maximum of the row and column sums and $M$ is the sum of all entries of the matrix, our algorithm runs in time linear in $M$ in expectation. Most previously published algorithms for this problem are approximate samplers based on Markov chain Monte Carlo, whose provable bounds on the mixing time are typically polynomials with rather large degrees.

math.CO↗