论文标题
不均匀的Erdős-rényi随机图的大偏差原理
The large deviation principle for inhomogeneous Erdős-Rényi random graphs
论文作者
论文摘要
考虑在$ n $顶点上的不均匀erdős-rényi随机图（errg），每个$ n $ i，j \ in \ in \ {1，\ ldots，n \} $，$ i \ neq j $均通过概率$ r_n（\ frac} wrac {i-n} n}，\ frac}，\ frac} { $（r_n）_ {n \ in \ mathbb {n}} $是一系列graphon，收敛到参考graphon $ r $。作为Chatterjee和Varadhan（2010年）对著名的Errgs的概括，Dhara and Sen（2019）证明了一系列图形的较大偏差原理（LDP），假设$ r $从0和1界限为$ r $，并且从0和1界限，并且具有下部半连接型的速率功能。我们进一步扩展了达拉（Dhara）和参议员的结果。我们将参考图的条件放松到$ \ log r，\ log（1-r）\ in l^1（[0,1]^2）$。我们还表明，在这种情况下，它们的速率函数等于不同，更可行的速率函数。然后，我们将这些结果应用于最大的不均匀差异特征值的大偏差原理，并削弱了Chakrabarty，Hazra，Den Hollander和Sfragara（2020）对速率函数分析的部分条件。
Consider the inhomogeneous Erdős-Rényi random graph (ERRG) on $n$ vertices for which each pair $i,j\in\{1,\ldots,n\}$, $i\neq j$ is connected independently by an edge with probability $r_n(\frac{i-1}{n},\frac{j-1}{n})$, where $(r_n)_{n\in\mathbb{N}}$ is a sequence of graphons converging to a reference graphon $r$. As a generalization of the celebrated LDP for ERRGs by Chatterjee and Varadhan (2010), Dhara and Sen (2019) proved a large deviation principle (LDP) for a sequence of such graphs under the assumption that $r$ is bounded away from 0 and 1, and with a rate function in the form of a lower semi-continuous envelope. We further extend the results by Dhara and Sen. We relax the conditions on the reference graphon to $\log r,\log(1-r)\in L^1([0,1]^2)$. We also show that, under this condition, their rate function equals a different, more tractable rate function. We then apply these results to the large deviation principle for the largest eigenvalue of inhomogeneous ERRGs and weaken the conditions for part of the analysis of the rate function by Chakrabarty, Hazra, Den Hollander and Sfragara (2020).
