Continual diagrams, Russian profiles, and the $\sqrt{n}$-scaling of a Young diagram

A continual diagram is a function $\omega :\mathbb{R}\to \mathbb{R}$ such that $\mathrm{\mid }\omega (x)-\omega (y)\mathrm{\mid }\le \mathrm{\mid }x-y\mathrm{\mid }$ for all real $x,y$ (the Lipschitz condition) and $\omega (x)=\mathrm{\mid }x\mathrm{\mid }$ for all sufficiently large $\mathrm{\mid }x\mathrm{\mid }$; the set of continual diagrams is denoted $D_0$. For $\omega \in D_0$ set ${\sigma }_{\omega }(x):=\frac{1}{2}(\omega (x)-\mathrm{\mid }x\mathrm{\mid })\mathrm{.}$ Since $\omega$ agrees with $\mathrm{\mid }\cdot \mathrm{\mid }$ outside a compact set, ${\sigma }_{\omega }$ is compactly supported; and ${\sigma }_{\omega }$ is 1-Lipschitz, because $\mathrm{\mid }{\sigma }_{\omega }(x)-{\sigma }_{\omega }(y)\mathrm{\mid }=\frac{1}{2}\mathrm{\mid }\omega (x)-\omega (y)-(\mathrm{\mid }x\mathrm{\mid }-\mathrm{\mid }y\mathrm{\mid })\mathrm{\mid }\le \frac{1}{2}(\mathrm{\mid }x-y\mathrm{\mid }+\mathrm{\mid }x-y\mathrm{\mid })$ by the Lipschitz bound and $\mid \mathrm{\mid }x\mathrm{\mid }-\mathrm{\mid }y\mathrm{\mid }\mid \le \mathrm{\mid }x-y\mathrm{\mid }$: the latter follows by applying The triangle inequality to $x=(x-y)+y$ and to $y=(y-x)+x$, with Basic properties of the absolute value. For $s>0$ define the $s$-scaling ${\omega }_s(x):=s^{-1}\omega (sx)$. Then ${\omega }_s\in D_0$, and ${\sigma }_{{\omega }_s}(x)=s^{-1}{\sigma }_{\omega }(sx),x\in \mathbb{R},$ directly from the definition, so scaling preserves the Lipschitz constant.

The empty partition has profile $\mathrm{\varnothing }(x):=\mathrm{\mid }x\mathrm{\mid }$. Every nonempty $\lambda ⊢n$ determines a continual diagram $\lambda (\cdot )\in D_0$ (Partitions, English diagrams, and conjugation). Regard the boxes of $[\lambda ]$ as the unit squares $[i-1,i]\times [j-1,j]$, $1\le i\le k$, $1\le j\le {\lambda }_i$, in the plane with coordinates $(r,s)$, and rotate by $x=s-r$, $y=r+s$. The outer staircase of the image, extended by the two axis rays, is the graph of a continuous piecewise linear function $y=\lambda (x)$, with ${\lambda }^{\mathrm{\prime }}(x)=\pm 1$ at every point where the derivative exists (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set): each horizontal or vertical unit step of the boundary staircase becomes a unit step of slope $\mp 1$ under the linear map $(r,s)\mapsto (s-r,r+s)$, and the graph is read from the outer corner at $x=-{\lambda }_1^{\mathrm{\prime }}$ to the corner at $x={\lambda }_1$. Outside these corners the boundary follows the axis strip, so $\lambda (x)=\mathrm{\mid }x\mathrm{\mid }\text{for }x\ge {\lambda }_1\text{and for }x\le -{\lambda }_1^{\mathrm{\prime }},$ the extreme corners being the end of the first row and the bottom of the first column. Hence $\lambda (\cdot )\in D_0$ and the support of ${\sigma }_{\lambda }$ is contained in $[-{\lambda }_1^{\mathrm{\prime }},{\lambda }_1]$. The area identity, also valid for the empty profile, holds: $\frac{1}{2}{\int }_{\mathbb{R}}(\lambda (x)-\mathrm{\mid }x\mathrm{\mid })dx=n,$ because the compact region $\{(x,y):-\lambda {\mathrm{\prime }}_1\le x\le {\lambda }_1,\mathrm{\mid }x\mathrm{\mid }\le y\le \lambda (x)\}$ is exactly the image under the invertible linear map $T(r,s):=(s-r,r+s)$, of determinant $-2$, of the union $[\lambda ]$ of the $n$ unit squares, and the image of a Jordan-measurable compact set of area $n$ has area $\mathrm{\mid }\det T\mathrm{\mid }\cdot n=2n$ (Change of variables for an injective $C^1$ map on a compact Jordan set, applied with $f\equiv 1$); the region is the area under the continuous piecewise linear function $\lambda (x)-\mathrm{\mid }x\mathrm{\mid }\ge 0$ over the compact interval $[-{\lambda }_1^{\mathrm{\prime }},{\lambda }_1]$, which is its Riemann-Darboux integral (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$).

The $\sqrt{n}$-scaled profile of $\lambda ⊢n$, $n\ge 1$, is $\bar{\lambda }(x):=n^{-1\mathrm{/}2}\lambda \left(n^{1\mathrm{/}2}x\right)={\lambda }_{n^{1\mathrm{/}2}}(x),x\in \mathbb{R},$ the $\sqrt{n}$-scaling of the preceding paragraph. Thus $\bar{\lambda }\in D_0$, $\bar{\lambda }(x)=\mathrm{\mid }x\mathrm{\mid }$ for $\mathrm{\mid }x\mathrm{\mid }\ge \max ({\lambda }_1,{\lambda }_1^{\mathrm{\prime }})\mathrm{/}\sqrt{n}$, and ${\sigma }_{\bar{\lambda }}(x)=n^{-1\mathrm{/}2}{\sigma }_{\lambda }(n^{1\mathrm{/}2}x)$; the area identity scales to $\frac{1}{2}{\int }_{\mathbb{R}}(\bar{\lambda }(x)-\mathrm{\mid }x\mathrm{\mid })dx=1$. The probability distribution with which $\lambda$ is drawn in this batch is the Plancherel measure of The Plancherel measure on the partitions of $n$. No choice principle is used.