Hi!
Does anyone know of any statement relating the probability that the second largest eigenvalue of a random graph is bigger than $x$ to the parameter $p$ in the $G_{np}$ model?
Hi! Does anyone know of any statement relating the probability that the second largest eigenvalue of a random graph is bigger than $x$ to the parameter $p$ in the $G_{np}$ model? 


Provided $pN$ is large in comparison to $N^{2/3}$ it looks like this paper by László Erdős, Antti Knowles, HorngTzer Yau, and Jun Yin has what you need. Here's the abstract:
Skimming their introduction it seems that the distribution in question is the TracyWidom distribution. They also say that this result is expected to hold for $pN\gg1$, citing private communication by P. Sarnak. More explicitly, theorem 2.7 of the paper in your notation says the following. Let $A$ be the adjacency matrix of $G_{n,p}$ multiplied by the factor $\gamma/q=\frac{1}{\sqrt{(1p)pn}}$ where $p=q^2/n$ and $\gamma=(1q^2/n)^{1/2}$. Let $\mu_{n1}$ be the second largest eigenvalue of $A$ (note that they index from the smallest to largest). Suppose that $q\geq n^\phi$ with $1/3<\phi\leq 1/2$. Then there exists $\delta>0$ such that for any $s$:
Here $F_1(s)$ is the cumulative distribution function for the TracyWidom distribution for the orthogonal ensemble. The Wikipedia article I linked above has formulas for $F_1(s)$. Because of the scale factor in front of $A$, their $\mu_{n1}$ should be related to your $\lambda_2$ by the factor $\gamma/q=\frac{1}{\sqrt{(1p)pn}}$. Thus after a bit of algebra, for large $n$: $PR\left[\lambda_2\leq (s n^{2/3}+2)\sqrt{(1p)pn}\right]\approx F_1(s)$ Letting $x=(sn^{2/3}+2)\sqrt{(1p)pn}$, and noting that you want the probability that $\lambda_2$ is greater than $x$, if I haven't made any mistakes in transcription:


