arXiv subjects

Jing-Yu Xu

Publications and source records attributed to Jing-Yu Xu.

6 recordsLinked to original sources

arXiv2026

Critical fluctuations in the edge--triangle exponential random graph models

We prove quantitative limit theorems for the edge density $Q$ in the dense edge--triangle exponential random graph model at criticality and under parameter perturbations of order $n^{-1}$. At the fixed critical parameters $(α_c,h_c),$ the centered fluctuations satisfy $n^{2/3}(Q-\mathbb{E} Q)\Rightarrow \mathcal{N}(0,σ_0^2)$, with explicit variance $σ_0^2>0$. This disproves the quartic fluctuation conjecture of Bianchi, Collet and Magnanini (2024). Gaussian limits also hold $(α_c+ λ/n,h_c + γ/n)$, except when the coefficients satisfy $-\frac{11}{16}+\frac{2}{9}(γ+\frac49λ)=0$. Under this linearity condition, the limiting laws of $\sqrt n(Q-2/3)$ have densities proportional to $\exp\{-81z^4/64+9θz^2/8\}$, where $θ=-57/16+8λ/27$. We obtain a sharp $n^{-1/3}$ rate in $1$-Wasserstein distance for the Gaussian limits and an $O(n^{-1/2})$ bound for the quartic limits. The proofs combine Stein's method for exchangeable pairs with conditional row estimates and a local Dobrushin contraction.

math.PR↗

arXiv2026

Three quantitative central limit theorems for the unbounded excursion component of a Gaussian field

For a class of smooth stationary Gaussian fields, we study the fluctuations of the volume, surface area and Euler characteristic of the unbounded excursion component in growing boxes $Λ_n=[-n,n]^d$. We establish nondegenerate central limit theorems for all three functionals throughout the supercritical regime in every dimension $d\geq 2$. We further quantify these central limit theorems by establishing Berry--Esseen bounds of order $O\bigl(|Λ_n|^{-(1-1/s)}\bigr)$ in Kolmogorov distance, where $s\in(1,2]$ depend on the regularity and decay assumptions. Especially for the Bargmann--Fock field, we obtain a Berry--Esseen bound of $O\bigl(|Λ_n|^{-1/2}\bigr)$ for all three functionals. A matching lower bound shows that this rate is sharp for the Euler characteristic. For the volume, we improve the previously known rate of $O\bigl(|Λ_n|^{-1/2}(\log|Λ_n|)^{2d}\bigr)$. We mainly use Stein's method and moment bounds for differences arising from white-noise resampling to finish our proof.

math.PR↗

arXiv2026

Normal approximation for total magnetization in subcritical long-range Ising models

We establish Berry--Esseen bounds and 1-Wassershtein bounds for the total spin of strictly subcritical zero-field Ising models with summable interactions. Let $J:\mathbb Z^d\to[0,\infty)$ be symmetric, satisfy $J(0)=0$, and $\|J\|_1:=\sum_{x\in\mathbb Z^d}J(x)<\infty$. For a nonempty finite set $Λ\subset\mathbb Z^d$, consider zero-field Gibbs measure with free boundary conditions, in which a spin configuration $σ\in\{-1,+1\}^Λ$ has probability proportional to \[ \exp\!\left\{ β\sum_{x,y\inΛ} J(x-y)σ_xσ_y \right\}. \] Let $S_Λ:=\sum_{x\inΛ}σ_x$ be the total spin and $β_{\mathrm c}$ denote the critical inverse temperature. For every fixed $0\leqβ<β_{\mathrm c}$, we prove that \[ d_{\mathrm K}\!\left( \frac{S_Λ}{\sqrt{\operatorname{Var}(S_Λ)}},G \right) \leq C_{β,J}|Λ|^{-1/2}, \] where $d_{\mathrm K}$ is the Kolmogorov distance, $G\sim \mathcal{N}(0,1)$, and $C_{β,J}$ depends only on $β$ and $J$. For the fully periodized model on $(\mathbb Z/L\mathbb Z)^d$, the corresponding rate is $L^{-d/2}$, uniformly in $L$. For the classical nearest-neighbor model, the Kolmogorov bound also holds for $-β_{\mathrm c}<β<β_{\mathrm c}$, covering both ferromagnetic and antiferromagnetic interactions. We also obtain $1$-Wasserstein bounds with the same rates in the respective subcritical regimes. Both the Kolmogorov and $1$-Wasserstein bounds hold with the same rate under the corresponding infinite-volume free-boundary Gibbs measures.

math.PR↗

arXiv2026

Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums

Let $X_1,X_2,\ldots$ be i.i.d. random variables with zero mean and variance $σ^2\in(0,\infty)$. In this paper, we establish a sharp Cramér-type moderate deviation for the self-normalized maximum of partial sums under a finite $p$-th absolute moment, $2<p\le3$. In particular, we show that $\mathbb{P}(M_n\ge xV_n)=2(1-Φ(x))\{1+O(1)β_{n,p}(1+x)^p\}$ for $0\le x\leβ_{n,p}^{-1/p}$, where $M_n=\max_{1\le k\le n}\sum_{j=1}^kX_j$, $V_n=(\sum_{j=1}^nX_j^2)^{1/2}$, $β_{n,p}=\mathbb{E}|X_1|^p/(n^{(p-2)/2}σ^p)$, and $Φ$ is the standard normal distribution function. This proves a conjecture of Shao2010. More precisely, we bound the relative error by the same truncated-moment quantity as for self-normalized sums. The proof compares the maximum with the absolute sum while preserving their common denominator, through a reflected-walk representation and Fourier estimates with distribution-adapted smoothing.

math.PR↗

arXiv2026

The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants

Let $A=(A_n)_{n\ge2}$ be a triangular array of random matrices, where $A_n=(a_{ij})_{1\le i,j\le n}$ is an $n\times n$ random matrix with independent real entries satisfying $\mathbb E a_{ij}=0$ and $\mathbb Ea_{ij}^2=1$, and put $\mathcal L_n=\log|\det A_n|$ and \[ W_n^{\mathrm d}(A_n):=\frac{\mathcal L_n - \frac12\log(n-1)!}{\sqrt{\frac12\log n}},\quad W_n^{\mathrm e}(A_n):= \frac{\mathcal L_n-\mathbb E \mathcal L_n}{\sqrt{\frac12\log n}}. \] We prove that $W_n^{\mathrm d}(A_n) \Rightarrow \mathcal N(0,1)$, whenever the family $\left\{\frac{|a_{ij}|^{4}}{\sqrt{\log(e+|a_{ij}|)}} \right\}_{n\geq 2;1\leq i,j\leq n}$ is uniformly integrable. If, in addition, the entries have uniformly bounded densities, then $W_n^{\mathrm e}(A_n) \Rightarrow \mathcal N(0,1)$ whenever the family $\left\{\frac{|a_{ij}|^{4}}{\log(e+|a_{ij}|)}\right\}_{n\geq 2;1\leq i,j\leq n} $ is uniformly integrable. These two conditions are optimal at the level of universal moment assumptions. We further establish the corresponding Berry--Esseen bounds, and show that for $0<δ\le\tfrac12$, if $\sup\limits_{n}\max\limits_{1\le i,j\le n}\mathbb E \frac{|a_{ij}|^4} {\{\log(e+|a_{ij}|)\}^{1/2-δ}}<\infty$, then \begin{align*} d_{\mathrm K}(W_n^{\mathrm d}(A_n),\mathcal N(0,1))\le C(\log n)^{-δ}. \end{align*} For $0<γ\le1$, if $\sup\limits_{n}\max\limits_{1\le i,j\le n}\mathbb E \frac{|a_{ij}|^4} {\{\log(e+|a_{ij}|)\}^{1-γ}}<\infty$ and the entries have uniformly bounded densities, then \begin{align*} d_{\mathrm K}(W_n^{\mathrm e}(A_n),\mathcal N(0,1))\le C(\log n)^{-γ}. \end{align*} When $δ= 1/2$ and $γ= 1$, the bounds $(\log n)^{-1/2}$ and $(\log n)^{-1}$ are optimal, respectively. Our results improve the earlier Central Limit Theorem by \cite{BaoPanZhou2015} and the Berry--Esseen bound by \cite{NguyenVu2014}.

math.PR↗

arXiv2026

Critical-Window Fluctuations and Disorder Universality for the Sherrington--Kirkpatrick Model

We establish free-energy fluctuation limits for the Ising Sherrington--Kirkpatrick model in the nonzero parts of its critical window. For fixed $b\ne0$ and $β_N=1+bN^{-1/3}\sqrt{\log N}$, our main Gaussian orthogonal ensemble (GOE) result is \[ \sqrt{\frac6{\log N}}\left(F_{N,β_N}-N\,\mathrm{FE}(β_N)+\frac{\log N}{12}\right)\xrightarrow{d}G+\sqrt{\frac32}\,b_+TW_1, \] where $G$ is standard Gaussian, $TW_1$ has the real Tracy--Widom law, and $G$ is independent of $TW_1$, and $\mathrm{FE}$ denotes the limit of spherical Sherrington--Kirkpatrick free-energy. Additionally, we show that in a moderately supercritical regime \[ \frac{2}{N^{1/3}(β_N-1)}\left(F_{N,β_N}-N\,\mathrm{FE}(β_N)+\frac{\log N}{12}\right)\xrightarrow{d}TW_1. \] We also show that the above results remains valid for independent, not necessarily identically distributed, disorder matrices whose first three moments match the Gaussian law and whose fourth moments satisfy an averaged bound.

math.PR↗