Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs

We consider an inhomogeneous Erdős-Rényi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $λ_N$ be the maximal eigenvalue of the adjacency matrix of $G_N$. It is known that $λ_N/N$ satisfies a large deviation principle as $N \to \infty$. The associated rate function $ψ_r$ is given by a variational formula that involves the rate function $I_r$ of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of $ψ_r$, specially when the reference graphon is of rank 1.

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