Question

This is a request for reference. Where can I find discussion of the Erdős–Rényi random graph in the regime between "giant-component" and "fully connected"? e.g. a detailed picture for say, $p_n=\frac{(\log n)^\gamma}{n}$, with $0<\gamma<1$

Answer 1

This may not satisfy your needs, but this recent (Fall 2012) paper, or its references might help:

"Lack of Hyperbolicity in Asymptotic Erdös-Renyi Sparse Random Graphs," by Onuttom Narayan, Iraj Saniee, Gabriel H. Tucci. (arXiv link).

Among other results, they have a new proof that

the giant component of $G(n,c/n)$ when $c>1$ has zero spectral gap almost surely as $n\to\infty$.

Random graphs in the $p = c/n$ (middle) regime are not $\delta$-hyperbolic [in the sense of Gromov], in the sense that they contain $\delta$-fat triangles for arbitrary large $\delta$, almost surely as $n\to\infty$:

 

Answer 2

"Random Graphs" by Bollobas has an extensive discussion of the thresholds at which various properties appear in the Erdos-Renyi model, for example the appearance of the first cycles. It's quite well-written, too. I'm familiar with the first edition, which is a little behind the times, but I see a new edition was published in 2001, which should be fairly up-to-date (the field has cooled down in the last decade).