AI 中文总结
研究Pittel无顶点增长优先连接过程中最大分量大小，在亚临界区域给出精确渐近公式，常数随参数变化并趋于Erdős–Rényi值。
AI 中文摘要
我们研究了Pittel无顶点增长优先连接过程中最大分量的大小。从固定顶点集$[n]$上的空图开始，边以正比于$(d_u+\alpha)(d_v+\alpha)$的概率逐条添加，其中$d_u$和$d_v$是$u$和$v$的当前度数，$\alpha>0$。令$L_1$表示最大分量的大小，并设$m_c:=\frac{\alpha n}{2(\alpha+1)}$。我们证明，如果$m=m_c(1-\varepsilon)$，$\varepsilon=\varepsilon(n)\to0$，$\varepsilon^3 n\to\infty$，那么对于每个固定的$\alpha>0$，有\\[ L_1=(1+o_p(1))\frac{2(\alpha+2)}{\alpha+1}\varepsilon^{-2}\log(\varepsilon^3 n)。\\] 更一般地，当$\alpha=\alpha(n)\to a\in(0,\infty]$时，相同的渐近性成立。特别地，常数$2(\alpha+2)/(\alpha+1)$随着$\alpha\to\infty$收敛到Erdős–Rényi值$2$。此外，如果$m=\left\lfloor \frac n2(1-\varepsilon)\right\rfloor$且$\alpha\varepsilon\to\infty$，那么\\[ L_1=(2+o_p(1))\varepsilon^{-2}\log(\varepsilon^3 n)。\\] $L_1$的亚临界渐近性解决了Janson和Warnke留下的问题。上界论证依赖于观察到：在对度数序列进行条件化后，图可以通过相应的配置模型处理；下界则来自树分量渐近性和二阶矩论证。
英文摘要
We study the size of the largest component in Pittel's preferential attachment process without vertex growth. Starting from the empty graph on a fixed vertex set $[n]$, edges are added one by one with probabilities proportional to $(d_u+α)(d_v+α)$, where $d_u$ and $d_v$ are the current degrees of $u$ and $v$, and $α>0$. Let $L_1$ denote the size of the largest component, and set $m_c:=\frac{αn}{2(α+1)}.$ We prove that if $m=m_c(1-\varepsilon), \varepsilon=\varepsilon(n)\to0, \varepsilon^3 n\to\infty,$ then \[ L_1=(1+o_p(1))\frac{2(α+2)}{α+1}\varepsilon^{-2}\log(\varepsilon^3 n) \] for every fixed $α>0$. Moreover, the same asymptotic holds whenever $α=α(n)\to a\in(0,\infty]$. In particular, the constant $2(α+2)/(α+1)$ converges to the Erdős--Rényi value $2$ as $α\to\infty$. If $m=\left\lfloor \frac n2(1-\varepsilon)\right\rfloor$ and $α\varepsilon\to\infty$, then \[ L_1=(2+o_p(1))\varepsilon^{-2}\log(\varepsilon^3 n). \] The subcritical asymptotics for \(L_1\) resolve the problem left open by Janson and Warnke. The upper bound argument relies on the fact that, after conditioning on the degree sequence, the graph can be treated through the corresponding configuration model, the lower bound follows from tree component asymptotics and a second moment argument.
