The Logan-Shepp-Vershik-Kerov limit profile $\mathrm{\Omega }$

Define the function $\mathrm{\Omega }:\mathbb{R}\to \mathbb{R}$ by $\mathrm{\Omega }(x)=\frac{2}{\pi }(x\arcsin \frac{x}{2}+\sqrt{4-x^2})(\mathrm{\mid }x\mathrm{\mid }\le 2),\mathrm{\Omega }(x)=\mathrm{\mid }x\mathrm{\mid }(\mathrm{\mid }x\mathrm{\mid }\ge 2),$ with the principal arcsine of Principal inverse sine and inverse cosine. Its elementary properties, all used below, are as follows.

(a) Evenness. The functions $x\mapsto x\arcsin (x\mathrm{/}2)$, $x\mapsto \sqrt{4-x^2}$ and $x\mapsto \mathrm{\mid }x\mathrm{\mid }$ are even, so $\mathrm{\Omega }$ is even.

(b) Values and continuity at the junctions. At $x=\pm 2$ the first formula gives $\frac{2}{\pi }(\pm 2\arcsin (\pm 1)+0)=2$ because $\arcsin (\pm 1)=\pm \pi \mathrm{/}2$, agreeing with $\mathrm{\mid }x\mathrm{\mid }=2$; the arcsine branch and $\mathrm{\mid }x\mathrm{\mid }$ are continuous on their closed domains, and the two branches agree at the two junction points, so $\mathrm{\Omega }$ is continuous on all of $\mathbb{R}$.

(c) First derivative. For $\mathrm{\mid }x\mathrm{\mid }<2$ differentiability of $\arcsin$ on $(-1,1)$ (Principal inverse sine and inverse cosine, Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c\in I$ with $f^{\mathrm{\prime }}(c)\mathrm{\ne }0$, then the inverse $g$ is differentiable at $f(c)$ with $g^{\mathrm{\prime }}(f(c))=1\mathrm{/}{f}^{\mathrm{\prime }}(c)$; and if $f^{\mathrm{\prime }}(c)=0$ then $g$ is not differentiable at $f(c)$, The derivatives of sine and cosine are cosine and minus sine) and the chain and product rules (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$) applied to $x\mapsto \frac{2}{\pi }(x\arcsin \frac{x}{2}+\sqrt{4-x^2})$ give ${\mathrm{\Omega }}^{\mathrm{\prime }}(x)=\frac{2}{\pi }(\arcsin \frac{x}{2}+\frac{x}{2}(1-\frac{x^2}{4}{)}^{-1\mathrm{/}2}-\frac{x}{2}(1-\frac{x^2}{4}{)}^{-1\mathrm{/}2})=\frac{2}{\pi }\arcsin \frac{x}{2}\mathrm{.}$ For $\mathrm{\mid }x\mathrm{\mid }>2$ the derivative of the restriction is ${\mathrm{\Omega }}^{\mathrm{\prime }}(x)=\mathrm{sgn}x$. As $x\to 2^{-}$ the formula tends to $\frac{2}{\pi }\cdot \frac{\pi }{2}=1=\mathrm{sgn}x$ for $x>2$, and as $x\to -2^{+}$ it tends to $-1=\mathrm{sgn}x$ for $x<-2$; hence $\mathrm{\Omega }$ is differentiable at every real point with ${\mathrm{\Omega }}^{\mathrm{\prime }}(x)=\frac{2}{\pi }\arcsin \frac{x}{2}(\mathrm{\mid }x\mathrm{\mid }<2),{\mathrm{\Omega }}^{\mathrm{\prime }}(x)=\mathrm{sgn}x(\mathrm{\mid }x\mathrm{\mid }>2),$ and the one-sided derivatives at $x=\pm 2$ both equal $\pm 1$ (they are the limits of ${\mathrm{\Omega }}^{\mathrm{\prime }}$ from within $(-2,2)$ and from outside).

(d) Lipschitz bound and smoothness. Since $\mathrm{\mid }\arcsin y\mathrm{\mid }<\pi \mathrm{/}2$ for $\mathrm{\mid }y\mathrm{\mid }<1$, one has $\mathrm{\mid }{\mathrm{\Omega }}^{\mathrm{\prime }}(x)\mathrm{\mid }<1$ for $\mathrm{\mid }x\mathrm{\mid }<2$, while $\mathrm{\mid }{\mathrm{\Omega }}^{\mathrm{\prime }}(x)\mathrm{\mid }=1$ for $\mathrm{\mid }x\mathrm{\mid }>2$. On each of the intervals $(-\mathrm{\infty },-2]$, $[-2,2]$, $[2,\mathrm{\infty })$ the function is continuous and differentiable on the interior, so the mean value theorem (The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$) gives $\mathrm{\mid }\mathrm{\Omega }(x)-\mathrm{\Omega }(y)\mathrm{\mid }\le \mathrm{\mid }x-y\mathrm{\mid }$ for $x,y$ in the same interval, and the continuity at $\pm 2$ gives the same bound across the junctions; thus $\mathrm{\Omega }$ is $1$-Lipschitz. On $(-2,2)$ the arcsine branch is $C^{\mathrm{\infty }}$ with ${\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}(x)=\frac{2}{\pi }(4-x^2{)}^{-1\mathrm{/}2}>0,$ so $\mathrm{\Omega }$ is $C^{\mathrm{\infty }}$ there with ${\mathrm{\Omega }}^{\mathrm{\prime }}$ strictly increasing. Since $\mathrm{\mid }\mathrm{\Omega }(x)-\mathrm{\Omega }(y)\mathrm{\mid }\le \mathrm{\mid }x-y\mathrm{\mid }$ for all $x,y$ and $\mathrm{\Omega }(x)=\mathrm{\mid }x\mathrm{\mid }$ for $\mathrm{\mid }x\mathrm{\mid }\ge 2$, we have $\mathrm{\Omega }\in D_0$ in the sense of Continual diagrams, Russian profiles, and the $\sqrt{n}$-scaling of a Young diagram, with ${\sigma }_{\mathrm{\Omega }}=\frac{1}{2}(\mathrm{\Omega }-\mathrm{\mid }x\mathrm{\mid })$ supported in $[-2,2]$ and ${\sigma }_{\mathrm{\Omega }}(0)=2\mathrm{/}\pi >0$. No choice principle is used.