Author

Song-Hao Liu

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Preprint Sep 2026

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 $\beta_N=1+bN^{-1/3}\sqrt{\log N}$, our main Gaussian orthogonal ensemble (GOE) result is \[ \sqrt{\frac6{\log N}}\left(F_{N,\beta_N}-N\,\mathrm{FE}(\beta_N)+\frac{...

Yu Cheng, Song-Hao Liu, Qi-Man Shao et al. · 1 citation · ⚡1

Preprint Sep 2026

Mixing Time of Conditional Two Star Exponential Random Graphs

Two star exponential random graph models (ERGMs) are an interesting special case of both general ERGMs and mean-field Ising models. In this paper, we study two star ERGMs conditioning on the edge density $p$. We begin with an analytic characterization of the replica symmetric region, where the conditional model is clos...

Xiao Fang, Song-Hao Liu, Xiao-Lin Wang · 0 citations

Preprint Sep 2026

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\lo...

Song-Hao Liu, Qi-Man Shao, Jing-Yue Xu · 0 citations