In their landmark paper from 1977 named "A Variational Problem for Random Young Tableaux" Logan and Shepp obtained a number of results concerning asymptotic properties of Young Diagrams sampled with Plancherel law.

In the third and last section of the paper they study asymptotic shapes under restrictions on the value of first row and first column. For instance, for Young diagrams $\lambda \vdash n$, conditioned to $$ \lambda_1 \approx a \sqrt{n} \qquad \text{and} \qquad \lambda_1' \approx b \sqrt{n}, $$ they claim that the optimal shape $\lambda$ is described by the explicit function $f_0(x;a,b)$ given in the picture below.

Equations (3.8)–(3.17)

However these formulas for $f_0(x;a,b)$ seem problematic to me. For instance, setting $a=b=2$ one should recover the optimal shape $$ f_0(x) = \frac{2}{\pi}\big( \sin \theta - \theta \cos \theta \big), \qquad x = f_0(x) + 2 \cos \theta, \qquad \text{for } 0 \le \theta \le \pi, $$ but this does not seem to be the case. In fact the limiting form of $f_0(x;2,2)$ seems to be \begin{align} 2 (1-\cos (\theta ))+\frac{2 (\sin (\theta )+\theta \cos (\theta ))}{\pi } \\ {}+2 (\cos (\theta )-1) \tan ^{-1}\left(\frac{\sin (\theta )}{1-\cos (\theta )}\right) \\ {}-2 (\cos (\theta )+1) \tan ^{-1}\left(\frac{\sin (\theta )}{\cos (\theta )+1}\right) \end{align} as when $a=b=2$ we have $$ \alpha =1 \qquad \beta =1, \qquad A=\frac{1}{4}, \qquad c=1, \qquad d=1. $$

Maybe I am making some trivial mistake. Has anyone looked into this?



The correct formulas should be given setting \begin{equation} \begin{split} f_0(x;a,b) = \frac{d-\cos \theta}{\sqrt{A}} + \frac{1}{\pi \sqrt{A}} \bigg[ & \sin \theta + \theta (\alpha - \beta + \cos \theta ) \\ & + 2 (\cos \theta - d) \arctan \frac{\beta \sin \theta}{1 - \beta \cos \theta} \\ & - 2 (\cos \theta + c) \arctan \frac{\alpha \sin \theta}{1 + \alpha \cos \theta} \bigg], \end{split} \end{equation} keeping the notation consistent with that of Section 3 of Logan and Shepp's paper (as in the picture).

In other words the mistake in the formula showed in the picture seems to only be a factor $1/\pi \sqrt{A}$ missing from lines 2 and 3 of equation (3.16). It is interesting to notice that throughout Section 3 of the same paper there are many other typos, for instance when authors consider less general cases of the above formulas specializing $\beta =1$. Corrections to these further typos should follow specializing the above corrected formula for $f_0(x;a,b)$.

If I find time, I'll turn this update into an answer.

  • 4
    $\begingroup$ Not an answer, but presumably you are also aware of the contemporaneous work of Vershik and Kerov on this problem? Maybe their paper has the correct formulas... $\endgroup$ Nov 26, 2022 at 19:11
  • 3
    $\begingroup$ Thanks for the comment @SamHopkins. I don`t think they treat optimal shapes with restrctions, do they? They provide a complete proof of optimality of $f_0(x)$, which I am familiar with, but I don't know about more geral optimal shapes. $\endgroup$
    – Matteo
    Nov 26, 2022 at 19:20


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.