Let $d,n \to \infty$ with fixed $\log(n)/d \to \alpha$, for some fixed $\alpha>0$. Thus, both $n$ and $d$ are large by $n$ is exponentially larger than $d$. Let $g_1,...,g_n$ be iid from $N(0,1)$ and set $G_i := \sigma_i g_i$, for some bounded sequence of positive numbers $\sigma_1,\ldots,\sigma_n$. For any $\lambda>0$, define $$ N_n(\lambda):= \#\{i \in [n] \mid G_i \ge \lambda\sqrt d\} = \sum_{i=1}^n 1_{\{G_i \ge \lambda \sqrt d\}} $$
Question. What is the asymptotic behavior of $N_n(\lambda)$?
This can come in various forms, for example I'd like to find $\lambda_{crit} \in [0,\infty]$ such that $$ \frac{1}{d}\log N_n(\lambda) \to \begin{cases} s(\lambda),&\mbox{ if }\lambda < \lambda_{crit},\\ -\infty,&\mbox{ if }\lambda>\lambda_{crit}, \end{cases} $$ for some function $s:(0,\infty) \to \mathbb R$ such that $s(\lambda) > 0$ for all $\lambda < \lambda_{crit}$.
Special Case: Random Energy Model
Note that the "homogeneous" case $\sigma_k = \sigma$ for all $k$ is well understood as it corresponds to the so-called Random Energy Model (statistical physics). Indeed, in this case it is well-known that $$ \lambda_{crit} = \sigma\sqrt{2\alpha},\quad s(\lambda) = \frac{\lambda_{crit}^2-\lambda^2}{2\sigma^2}. $$ For example if $\sigma=1$ and $n=2^d$, we get well-known formula $\lambda_{crit} = \sqrt{2\log 2}$.
See Section 1.1.1 of this manuscript https://arxiv.org/abs/1412.0958.
