In this important and interesting paper, the authors again illustrate the power of the Deift-Zhou steepest descent method associated with Riemann-Hilbert problems. Moreover, there is a new feature: they show how the technique may be applied, with suitable modifications, even in the presence of singularities. Let $\alpha>0$ and $$w(x):= \exp(-|x|^\alpha),\quad \alpha>0,\ x\in\bbfR.$$ Then we may define orthonormal polynomials $$p_n(x)= \gamma_n x^n+\cdots,\quad \gamma_n> 0,$$ satisfying $$\int p_n p_m w= \delta_{mn},\quad m,n\ge 0.$$ The authors obtain very precise (Plancherel-Rotach) asymptotics for the orthonormal polynomials in all regions of the plane, asymptotic relations for $\gamma_n$ with error terms, the zeros of $p_n$, the spacing between successive zeros, and so on.
There are two remarkable features in this particular work: firstly, they can handle not just $\alpha$ an even positive integer, as was the case in their earlier papers, but also non-integer $\alpha$; secondly, the case $\alpha\le 1$ is particularly difficult and defied efforts at asymptotics using other methods, such as Bernstein-SzegĂ¶ techniques. So this is one case, where Riemann-Hilbert techniques not only yield more precise results, but also work for weights where other methods failed to give any asymptotics at all.