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 close in cut distance to the Erd\H{o}s--R\'enyi random graph $G(n,p)$. We prove that this region is twice as large as the corresponding region for the unconditional model. To study refined properties of the conditional model, we then analyze the global Kawasaki algorithm for sampling from it. Within the replica symmetric region, and under the additional condition that $4\beta p(1-p)<0.5$ when $|p-1/2|\lesssim 0.4632$, where $\beta$ is the model parameter, we prove metastable fast mixing of the Kawasaki algorithm. As corollaries, we obtain a weak Poincar\'e inequality and use it to deduce concentration inequalities of optimal order for conditional subgraph counts. The additional condition $4\beta p(1-p)<1/2$ if $|p-1/2|\lesssim 0.4632$ comes from our proof technique of contractive coupling. This bottleneck did not appear in previous studies applying the contractive coupling technique to unconditional ERGMs.
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 $0<\delta\le\tfrac12$, if $\sup\limits_{n}\max\limits_{1\le i,j\le n}\mathbb E \frac{|a_{ij}|^4} {\{\log(e+|a_{ij}|)\}^{1/2-\delta}}<\infty$, then \begin{align*} d_{\mathrm K}(W_n^{\mathrm d}(A_n),\mathcal N(0,1))\le C(\log n)^{-\delta}. \end{align*} For $0<\gamma\le1$, if $\sup\limits_{n}\max\limits_{1\le i,j\le n}\mathbb E \frac{|a_{ij}|^4} {\{\log(e+|a_{ij}|)\}^{1-\gamma}}<\infty$ 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)^{-\gamma}. \end{align*} When $\delta = 1/2$ and $\gamma = 1$, 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}.
