Reference asking: phase transition in SAT
The two most relevant rigorous results are:


Ehud Friedgut, Sharp thresholds of graph properties, and the $k$-SAT problem. This paper (with an appendix by Jean Bourgain) shows that $k$-SAT exhibits a sharp threshold. However, a priori this threshold could depend on $n$ (i.e., this method cannot show that $\alpha$ is constant).
Jian Ding, Allan Sly, Nike Sun, Proof of the satisfiability conjecture for large $k$. The authors determine the exact value of $\alpha$ for $k \geq k_0$. This value of $\alpha$ had been calculated by physicists using the cavity method, but their arguments are not rigorous.


Related work discusses the geometry of the solution space of $k$-SAT around various thresholds. See for example Dimitris Achlioptas, Amin Coja-Oghlan, Federico Ricci-Tersenghi, On the solution‐space geometry of random constraint satisfaction problems.