Large deviations for the largest eigenvalue of Gaussian networks with constant average degree

Large deviation behavior of the largest eigenvalue $\lambda_1$ of Gaussian networks (Erd\H{o}s-R\'enyi random graphs $\mathcal{G}_{n,p}$ with i.i.d. Gaussian weights on the edges) has been the topic of considerable interest. Recently in [6,30], a powerful approach was introduced based on tilting measures by suitable spherical integrals, particularly establishing a non-universal large deviation behavior for fixed $p<1$ compared to the standard Gaussian ($p=1$) case. The case when $p\to 0$ was however completely left open with one expecting the dense behavior to hold only until the average degree is logarithmic in $n$. In this article we focus on the case of constant average degree i.e., $p=\frac{d}{n}$. We prove the following results towards a precise understanding of the large deviation behavior in this setting. 1. (Upper tail probabilities): For $\delta>0,$ we pin down the exact exponent $\psi(\delta)$ such that $$\mathbb{P}(\lambda_1\ge \sqrt{2(1+\delta)\log n})=n^{-\psi(\delta)+o(1)}.$$ Further, we show that conditioned on the upper tail event, with high probability, a unique maximal clique emerges with a very precise $\delta$ dependent size (takes either one or two possible values) and the Gaussian weights are uniformly high in absolute value on the edges in the clique. Finally, we also prove an optimal localization result for the leading eigenvector, showing that it allocates most of its mass on the aforementioned clique which is spread uniformly across its vertices. 2. (Lower tail probabilities): The exact stretched exponential behavior of $\mathbb{P}(\lambda_1\le \sqrt{2(1-\delta)\log n})$ is also established. As an immediate corollary, we get $\lambda_1 \approx \sqrt{2 \log n}$ typically, a result that surprisingly appears to be new. A key ingredient is an extremal spectral theory for weighted graphs obtained via the classical Motzkin-Straus theorem.