and:

$$ \sup\{s \in \mathbb{R} : \tilde{\omega}_{\text{reg}_e}(\lambda)(s) \neq |s|\} \to s_{\alpha_e}(\Omega), $$

in probability under $P_1n$ as $n \to +\infty$,

**Proof.** Write $s_\alpha$ instead of $s_{\alpha_e}(\Omega)$. Note that by (3.21) we have $s_\alpha > 0$. Let $\varepsilon > 0$ and let $\eta \in (0, 1)$ such that:

$$ \eta \le \min \left( \frac{\varepsilon}{2}, \frac{2\varepsilon}{s_\alpha} \right). \qquad (6.14) $$

Let $n \in \mathbb{Z}_{\ge 1}$ and let $M_n$ as in the proof of Theorem 6.1, that is, satisfying (6.5) where $m^\pm$ is defined in (6.4). Let $\lambda \in M_n$ and define $\mu := \text{reg}_e(\lambda)$. Recall from (6.12) that for all $s \in \mathbb{R}$ we have:

$$ \tilde{\rho}_{\mu}^{+}(s) \le \Omega_{e}^{+\eta}(s). $$

Thus, if $\{i_1 < \dots < i_{r+1}\}$ are the inner corners of $\mu$, we deduce from Proposition 5.10 together with Lemma 3.5 and Corollary 3.22 that:

$$ -2 - \eta \le \frac{i_1}{\sqrt{n}} \quad \text{and} \quad \frac{i_{r+1}}{\sqrt{n}} \le s_{\alpha_e}(\Omega^{+\eta}). $$

By Lemma 3.34 we deduce that:

$$ -2 - \eta - \frac{i_1 - 1}{\sqrt{n}} \le \frac{i_1 - 1}{\sqrt{n}} \quad \text{and} \quad \frac{i_{r+1}}{\sqrt{n}} \le (1 + \frac{\eta}{2})s_{\alpha_e}. $$

By Proposition 5.7, we thus have:

$$ \tilde{\rho}_{\mu}^{+}(s) = |s|, \quad \text{if } s \le -2 - \eta - \frac{1}{\sqrt{n}} \text{ or } s \ge (1 + \frac{\eta}{2})s_{\alpha_e}. $$

Recalling (6.9b), for all $s \in \mathbb{R}$ we have $|s| \le \tilde{\omega}_{\mu}(s) \le \tilde{\rho}_{\mu}^{+}(s)$ thus we deduce that:

$$ \tilde{\omega}_{\mu}(s) = |s|, \quad \text{if } s \le -2 - \eta - \frac{1}{\sqrt{n}} \text{ or } s \ge (1 + \frac{\eta}{2})s_{\alpha_e}. $$

Finally, we have proved that:

$$ \inf\{s \in \mathbb{R} : \tilde{\omega}_{\mu}(s) \neq |s|\} \ge -2 - \eta - \frac{1}{\sqrt{n}}, \qquad (6.15a) $$

and:

$$ \sup\{s \in \mathbb{R} : \tilde{\omega}_{\mu}(s) \neq |s|\} \le (1 + \frac{\eta}{2})s_{\alpha_e}. \qquad (6.15b) $$

Similarly, from $\Omega_e^{-\eta}(s) \le \tilde{\rho}_{\mu}^{-}(s)$ for all $s \in \mathbb{R}$ we deduce that:

$$ \frac{i_1 - 1}{\sqrt{n}} \le -2 + \eta \quad \text{and} \quad (1 - \frac{\eta}{2})s_{\alpha_e} \le \frac{i_{r+1}}{\sqrt{n}}, $$

hence recalling from (6.9a) the inequality $\tilde{\rho}_{\mu}^{-}(s) \le \tilde{\omega}_{\mu}(s)$ for all $s \in \mathbb{R}$ and Proposition 5.10 we know that if $s \in (-2 + \eta + \frac{1}{\sqrt{n}}, (1 - \frac{\eta}{2})s_{\alpha_e})$ we have $|s| < \tilde{\rho}_{\mu}^{-}(s)$ thus $|s| < \tilde{\omega}_{\mu}(s)$ as well. We deduce that:

$$ \inf\{s \in \mathbb{R} : \tilde{\omega}_{\mu}(s) \neq |s|\} \le -2 + \eta + \frac{1}{\sqrt{n}}, \qquad (6.16a) $$

and:

$$ \sup\{s \in \mathbb{R} : \tilde{\omega}_{\mu}(s) \neq |s|\} \ge (1 - \frac{\eta}{2})s_{\alpha_e}. \qquad (6.16b) $$
Hence, from (6.15) and (6.16) we obtain:

$$|\inf\{s \in \mathbb{R} : \tilde{\omega}_{\mu}(s) \neq |s|\} + 2| \leq \eta + \frac{1}{\sqrt{n}},$$

and:

$$|\sup\{s \in \mathbb{R} : \tilde{\omega}_{\mu}(s) \neq |s|\} - s_{\alpha}| \leq \frac{\eta}{2}s_{\alpha}.$$

With $n \gg 0$ such that $\frac{1}{\sqrt{n}} \leq \frac{\epsilon}{2}$ and recalling (6.14) we have $\eta + \frac{1}{\sqrt{n}} \leq \epsilon$ and $\frac{\eta}{2}s_{\alpha} \leq \epsilon$, thus we obtain the desired convergences since $\text{Pl}_n(M_n) \to 1$ as $n \to \infty$. $\square$

Finally, we deduce the asymptotic behaviour of the first part and the first column.

**Corollary 6.17.** Let $e \geq 2$. Under the Plancherel measure $\text{Pl}_n$:

(i) the rescaled size $\frac{1}{\sqrt{n}}\text{reg}_e(\lambda)_1$ of the first row of $\text{reg}_e(\lambda)$ converges as $n \to +\infty$ in probability to 2;

(ii) the rescaled size $\frac{1}{\sqrt{n}}\text{reg}_e(\lambda)'_1$ of the first column of $\text{reg}_e(\lambda)$ converges as $n \to +\infty$ in probability to $\frac{2e}{\pi}\sin\frac{\pi}{e}$.

*Proof.* The first point is clear by Theorem 6.13. For the second one, by the same theorem it suffices to prove that the announced value is equal to $s_{\alpha} = s_{\alpha}(\Omega)$, with $\alpha = \alpha_e = 1 - 2e^{-1}$. By Corollary 3.22 applied for $f = \Omega$ we have:

$$s_{\alpha} = (1 - \alpha)^{-1} \left[ \Omega(\Omega'^{-1}(\alpha)) - \alpha \Omega'^{-1}(\alpha) \right].$$

Recall from Lemma 2.11 that $\Omega'(s) = \frac{2}{\pi}\arcsin(\frac{s}{2})$ for $s \in (-2, 2)$. Hence, for $s \in (-2, 2)$ and $t \in (-1, 1)$ we have $\Omega'(s) = t \iff \arcsin(\frac{s}{2}) = \frac{\pi t}{2} \iff s = 2\sin\frac{\pi t}{2}$ thus $\Omega'^{-1}(t) = 2\sin\frac{\pi t}{2}$. We thus have:

$$\begin{aligned}
\Omega'^{-1}(\alpha) &= 2 \sin \frac{\pi \alpha}{2} \\
&= 2 \sin \frac{\pi (1 - 2e^{-1})}{2} \\
&= 2 \sin \left( \frac{\pi}{2} - \frac{\pi}{e} \right) \\
&= 2 \cos \frac{\pi}{e}.
\end{aligned}$$

We obtain:

$$\begin{aligned}
s_{\alpha} &= \frac{e}{2} \left[ \Omega(\Omega'^{-1}(\alpha)) - \alpha \Omega'^{-1}(\alpha) \right] \\
&= \frac{e}{2} \left[ \Omega(2 \cos \frac{\pi}{e}) - 2\alpha \cos \frac{\pi}{e} \right].
\end{aligned}$$

Now we have, recalling the identity $\arcsin(\cos x) = \frac{\pi}{2} - x$ for $x \in [0, \pi]$:

$$\begin{aligned}
\Omega \left( 2 \cos \frac{\pi}{e} \right) &= \frac{2}{\pi} \left[ \arcsin \left( \cos \frac{\pi}{e} \right) 2 \cos \frac{\pi}{e} + \sqrt{4 - 4 \cos^2 \frac{\pi}{e}} \right] \\
&= \frac{2}{\pi} \left[ \left( \frac{\pi}{2} - \frac{\pi}{e} \right) 2 \cos \frac{\pi}{e} + 2 \sin \frac{\pi}{e} \right] \\
&= 2 \left( 1 - \frac{2}{e} \right) \cos \frac{\pi}{e} + \frac{4}{\pi} \sin \frac{\pi}{e} \\
&= 2\alpha \cos \frac{\pi}{e} + \frac{4}{\pi} \sin \frac{\pi}{e},
\end{aligned}$$

thus we finally obtain:

$$s_{\alpha} = \frac{2e}{\pi} \sin \frac{\pi}{e}. \qquad (6.18)$$

$\square$
Hence, from (6.15) and (6.16) we obtain:

$$|\inf\{s \in \mathbb{R} : \tilde{\omega}_{\mu}(s) \neq |s|\} + 2| \leq \eta + \frac{1}{\sqrt{n}},$$

and:

$$|\sup\{s \in \mathbb{R} : \tilde{\omega}_{\mu}(s) \neq |s|\} - s_{\alpha}| \leq \frac{\eta}{2}s_{\alpha}.$$

With $n \gg 0$ such that $\frac{1}{\sqrt{n}} \leq \frac{\epsilon}{2}$ and recalling (6.14) we have $\eta + \frac{1}{\sqrt{n}} \leq \epsilon$ and $\frac{\eta}{2}s_{\alpha} \leq \epsilon$, thus we obtain the desired convergences since $\text{Pl}_n(M_n) \to 1$ as $n \to \infty$. $\square$

Finally, we deduce the asymptotic behaviour of the first part and the first column.

**Corollary 6.17.** Let $e \geq 2$. Under the Plancherel measure $\text{Pl}_n$:

(i) the rescaled size $\frac{1}{\sqrt{n}}\text{reg}_e(\lambda)_1$ of the first row of $\text{reg}_e(\lambda)$ converges as $n \to +\infty$ in probability to 2;

(ii) the rescaled size $\frac{1}{\sqrt{n}}\text{reg}_e(\lambda)'_1$ of the first column of $\text{reg}_e(\lambda)$ converges as $n \to +\infty$ in probability to $\frac{2e}{\pi}\sin\frac{\pi}{e}$.

*Proof.* The first point is clear by Theorem 6.13. For the second one, by the same theorem it suffices to prove that the announced value is equal to $s_{\alpha} = s_{\alpha}(\Omega)$, with $\alpha = \alpha_e = 1 - 2e^{-1}$. By Corollary 3.22 applied for $f = \Omega$ we have:

$$s_{\alpha} = (1 - \alpha)^{-1} \left[ \Omega(\Omega'^{-1}(\alpha)) - \alpha \Omega'^{-1}(\alpha) \right].$$

Recall from Lemma 2.11 that $\Omega'(s) = \frac{2}{\pi}\arcsin(\frac{s}{2})$ for $s \in (-2, 2)$. Hence, for $s \in (-2, 2)$ and $t \in (-1, 1)$ we have $\Omega'(s) = t \iff \arcsin(\frac{s}{2}) = \frac{\pi t}{2} \iff s = 2\sin\frac{\pi t}{2}$ thus $\Omega'^{-1}(t) = 2\sin\frac{\pi t}{2}$. We thus have:

$$\begin{aligned}
\Omega'^{-1}(\alpha) &= 2 \sin \frac{\pi \alpha}{2} \\
&= 2 \sin \frac{\pi (1 - 2e^{-1})}{2} \\
&= 2 \sin \left( \frac{\pi}{2} - \frac{\pi}{e} \right) \\
&= 2 \cos \frac{\pi}{e}.
\end{aligned}$$

We obtain:

$$\begin{aligned}
s_{\alpha} &= \frac{e}{2} \left[ \Omega(\Omega'^{-1}(\alpha)) - \alpha \Omega'^{-1}(\alpha) \right] \\
&= \frac{e}{2} \left[ \Omega(2 \cos \frac{\pi}{e}) - 2\alpha \cos \frac{\pi}{e} \right].
\end{aligned}$$

Now we have, recalling the identity $\arcsin(\cos x) = \frac{\pi}{2} - x$ for $x \in [0, \pi]$:

$$\begin{aligned}
\Omega \left( 2 \cos \frac{\pi}{e} \right) &= \frac{2}{\pi} \left[ \arcsin \left( \cos \frac{\pi}{e} \right) 2 \cos \frac{\pi}{e} + \sqrt{4 - 4 \cos^2 \frac{\pi}{e}} \right] \\
&= \frac{2}{\pi} \left[ \left( \frac{\pi}{2} - \frac{\pi}{e} \right) 2 \cos \frac{\pi}{e} + 2 \sin \frac{\pi}{e} \right] \\
&= 2 \left( 1 - \frac{2}{e} \right) \cos \frac{\pi}{e} + \frac{4}{\pi} \sin \frac{\pi}{e} \\
&= 2\alpha \cos \frac{\pi}{e} + \frac{4}{\pi} \sin \frac{\pi}{e},
\end{aligned}$$

thus we finally obtain:

$$s_{\alpha} = \frac{2e}{\pi} \sin \frac{\pi}{e}. \qquad (6.18)$$

$\square$
*Example 6.19.* Here are approximations of the first values of the limit for $\frac{1}{\sqrt{n}}\mathrm{reg}_e(\lambda)'_1$:

<table><thead><tr><th>e</th><th>2e/π</th><th>sin π/e</th></tr></thead><tbody><tr><td>2</td><td>1.27</td><td></td></tr><tr><td>3</td><td>1.65</td><td></td></tr><tr><td>4</td><td>1.80</td><td></td></tr></tbody></table>

The reader can check that they match the corresponding values of Figures 1.3, 1.4 and 6.1.

## References

[Bl] W. BLASCHKE, *Über affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten*. Leipzig Berichte **69** 436–453 (1917).

[CCG] S. CAMPI, A. COLESANTI and P. GRONCHI, *Shaking Compact Sets*. Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry Vol. **42**(1) (2001) 123–136.

[FrLu] A. FREYER and E. LUCAS, *Interpolating between volume and lattice point enumerator with successive minima*. Monatshefte für Mathematik **198** 717–740 (2022).

[Gr] P. M. GRUBER, *Convex and Discrete Geometry*. Springer, Grundlehren der mathe- matischen Wissenschaften **336** (2007).

[Ja] G. JAMES, *On the decomposition matrices of the symmetric groups II*. J. Algebra **43** 45–54 (1976).

[JaKe] G. JAMES and A. KERBER, *The Representation Theory of the Symmetric Group*. Encyclopedia of mathematics and its applications **16** Addison-Wesley (1981).

[KeVe] S. V. KEROV and A. M. VERSHIK, *Asymptotics of the Plancherel measure on the symmetric group and the limiting form of Young tableaux*. Sov. Math. Dokl. **18** 527–531 (1977).

[KrPa] S. G. KRANTZ and H. R. PARKS, *The Geometry of Domains in Space*. Birkhäuser Advanced Texts, Basler Lehrbücher (1999).

[LoSh] B. F. LOGAN and L. A. SHEPP, *A variational problem for random Young tableaux*. Adv. Math. **26** 206–222 (1977).

[Ma] A. MATHAS, *Iwahori–Hecke algebras and Schur algebras of the Symmetric Group*. University Lecture Series **15**, American Mathematical Society (1999).

[Mi] D. MILLAN BERDASCO, *Generalised regularisation maps on partitions*. arXiv:2105.05550 (2021).

[Rom] D. ROMIK, *The Surprising Mathematics of Longest Increasing Subsequences*. Cambridge University Press (2015).

[Sa] B. E. SAGAN, *The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions*. Graduate Texts in Mathematics **203**, Springer (2000).

[SM] SAGEMATH, the Sage Mathematics Software System. The Sage Developers, https://www.sagemath.org (2020).
*Example 6.19.* Here are approximations of the first values of the limit for $\frac{1}{\sqrt{n}}\mathrm{reg}_e(\lambda)'_1$:

<table><thead><tr><th>e</th><th>2e/π</th><th>sin π/e</th></tr></thead><tbody><tr><td>2</td><td>1.27</td><td></td></tr><tr><td>3</td><td>1.65</td><td></td></tr><tr><td>4</td><td>1.80</td><td></td></tr></tbody></table>

The reader can check that they match the corresponding values of Figures 1.3, 1.4 and 6.1.

## References

[Bl] W. BLASCHKE, *Über affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten*. Leipzig Berichte **69** 436–453 (1917).

[CCG] S. CAMPI, A. COLESANTI and P. GRONCHI, *Shaking Compact Sets*. Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry Vol. **42**(1) (2001) 123–136.

[FrLu] A. FREYER and E. LUCAS, *Interpolating between volume and lattice point enumerator with successive minima*. Monatshefte für Mathematik **198** 717–740 (2022).

[Gr] P. M. GRUBER, *Convex and Discrete Geometry*. Springer, Grundlehren der mathe- matischen Wissenschaften **336** (2007).

[Ja] G. JAMES, *On the decomposition matrices of the symmetric groups II*. J. Algebra **43** 45–54 (1976).

[JaKe] G. JAMES and A. KERBER, *The Representation Theory of the Symmetric Group*. Encyclopedia of mathematics and its applications **16** Addison-Wesley (1981).

[KeVe] S. V. KEROV and A. M. VERSHIK, *Asymptotics of the Plancherel measure on the symmetric group and the limiting form of Young tableaux*. Sov. Math. Dokl. **18** 527–531 (1977).

[KrPa] S. G. KRANTZ and H. R. PARKS, *The Geometry of Domains in Space*. Birkhäuser Advanced Texts, Basler Lehrbücher (1999).

[LoSh] B. F. LOGAN and L. A. SHEPP, *A variational problem for random Young tableaux*. Adv. Math. **26** 206–222 (1977).

[Ma] A. MATHAS, *Iwahori–Hecke algebras and Schur algebras of the Symmetric Group*. University Lecture Series **15**, American Mathematical Society (1999).

[Mi] D. MILLAN BERDASCO, *Generalised regularisation maps on partitions*. arXiv:2105.05550 (2021).

[Rom] D. ROMIK, *The Surprising Mathematics of Longest Increasing Subsequences*. Cambridge University Press (2015).

[Sa] B. E. SAGAN, *The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions*. Graduate Texts in Mathematics **203**, Springer (2000).

[SM] SAGEMATH, the Sage Mathematics Software System. The Sage Developers, https://www.sagemath.org (2020).