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The profile moments of $Ω$ are central binomial coefficients

Statement

For the limit profile $Ω$ of The Logan-Shepp-Vershik-Kerov limit profile $Ω$ , ${\tilde{p}}_{2 m} [Ω] = \frac{(2 m)!}{m! m!} = (\binom{2 m}{m}) (m \geq 1), {\tilde{p}}_{k} [Ω] = 0 (k odd),$ and ${\tilde{p}}_{1} [Ω] = 0$ by the declaration of Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ .

Facts & Assumptions

Given: the even profile $Ω$ and its profile moments ${\tilde{p}}_{k} [ω] = k (k - 1) \int_{R} x^{k - 2} σ_{ω} (x) d x$ , $σ_{ω} = \frac{1}{2} (ω - ∣ x ∣)$ (The Logan-Shepp-Vershik-Kerov limit profile $Ω$ , Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ).

[F1]

$Ω$ is even, $Ω (x) = ∣ x ∣$ for $∣ x ∣ \geq 2$ , and for $∣ x ∣ < 2$ it is $C^{1}$ with $Ω^{'} (x) = \frac{2}{π} \arcsin \frac{x}{2}$ (The Logan-Shepp-Vershik-Kerov limit profile $Ω$ ). Hence $σ_{Ω} = \frac{1}{2} (Ω - ∣ x ∣)$ is even, vanishes outside $[- 2, 2]$ , and is continuous and piecewise $C^{1}$ on $[- 2, 0]$ and $[0, 2]$ , with $σ_{Ω} (\pm 2) = 0$ ; integrating by parts on the two pieces gives, for $k \geq 2$ , $\int_{R} x^{k - 1} σ_{Ω}^{'} (x) d x = - (k - 1) \int_{R} x^{k - 2} σ_{Ω} (x) d x$ (all boundary terms vanish: at $\pm 2$ because $σ_{Ω}$ vanishes there, at $0$ because $k - 1 \geq 1$ ), so $- k \int_{R} x^{k - 1} σ_{Ω}^{'} = k (k - 1) \int_{R} x^{k - 2} σ_{Ω} = {\tilde{p}}_{k} [Ω]$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ).

[F2]

Monotone change of variables: if $φ : [c, d] \to [a, b]$ is a monotone differentiable bijection with integrable derivative and $f$ is Riemann integrable on $[a, b]$ , then $\int_{a}^{b} f = \int_{c}^{d} (f \circ φ) ∣ φ^{'} ∣$ (Monotone change of variable for Riemann-integrable functions).

[F3]

Wallis integrals: $I_{2 m} = \int_{0}^{π / 2} \sin^{2 m} θ d θ = \frac{π}{2} \prod_{k = 1}^{m} \frac{2 k - 1}{2 k} = \frac{π}{2} \cdot \frac{(2 m - 1)!!}{(2 m)!!}$ for $m \geq 0$ (Wallis integrals satisfy the two-step recurrence, closed forms, and the adjacent-integral squeeze); and $(2 m)! = 2^{m} m! (2 m - 1)!!$ (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ).

[F4]

Arcsine: $\arcsin (\sin θ) = θ$ for $θ \in [0, π / 2]$ (Principal inverse sine and inverse cosine), and $\sin$ and $\cos$ have the usual derivatives (The derivatives of sine and cosine are cosine and minus sine).

[F5]

Integration by parts on a closed interval: if $u, v$ are differentiable on $[a, b]$ with integrable derivatives then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ).

Proof

technique · direct

1.1givenF1algebra

Odd moments vanish: $σ_{Ω}$ is even and supported in $[- 2, 2]$ by [F1], so for odd $k \geq 3$ the integrand $x^{k - 2} σ_{Ω} (x)$ is odd and its integral over the symmetric interval vanishes; hence ${\tilde{p}}_{k} [Ω] = 0$ for odd $k$ by the definition, and ${\tilde{p}}_{1} [Ω] = 0$ by the convention.

1.2givenF1algebra

Reduction for even moments: fix $m \geq 1$ and put $k = 2 m$ . By [F1] and the integration-by-parts form of the profile moment, ${\tilde{p}}_{2 m} [Ω] = - 2 m \int_{R} x^{2 m - 1} σ_{Ω}^{'} (x) d x = - m \int_{- 2}^{2} x^{2 m - 1} (\frac{2}{π} \arcsin \frac{x}{2} - sgn x) d x .$ The integrand is even (odd factor times the odd function $\frac{2}{π} \arcsin (x / 2) - sgn x$ ), so the integral equals $2 m \int_{0}^{2} x^{2 m - 1} (1 - \frac{2}{π} \arcsin \frac{x}{2}) d x = \int_{0}^{2} (1 - \frac{2}{π} \arcsin \frac{x}{2}) d (x^{2 m})$ .

2.1givenF2F4step 1.2algebra

Substitution $x = 2 \sin θ$ : by [F2] applied to the increasing bijection $θ \mapsto 2 \sin θ$ from $[0, π / 2]$ onto $[0, 2]$ (with $2 \cos θ \geq 0$ and $\arcsin (\sin θ) = θ$ by [F4]), the integral of step 1.2 equals $\int_{0}^{π / 2} (1 - \frac{2}{π} θ) \cdot 4 m 2^{2 m - 1} \sin^{2 m - 1} θ \cos θ d θ .$

3.1givenF3step 2.1algebra

Integration by parts and Wallis: on $[0, π / 2]$ put $u (θ) : = 1 - \frac{2}{π} θ$ and $v (θ) : = \frac{\sin^{2 m} θ}{2 m}$ ; both are differentiable with continuous derivatives $u^{'} \equiv - \frac{2}{π}$ and $v^{'} = \sin^{2 m - 1} θ \cos θ$ , so [F5] gives $\int_{0}^{π / 2} u v^{'} = [u v]_{0}^{π / 2} - \int_{0}^{π / 2} u^{'} v = \frac{2}{π} \cdot \frac{1}{2 m} \int_{0}^{π / 2} \sin^{2 m} θ d θ = \frac{I_{2 m}}{π m},$ because $u (π / 2) = 0$ and $u (0) = 1$ while $v (0) = \sin 0 = 0$ . Multiplying by $4 m 2^{2 m - 1}$ and using [F3], ${\tilde{p}}_{2 m} [Ω] = \frac{2^{2 m + 1}}{π} I_{2 m} = \frac{2^{2 m + 1}}{π} \cdot \frac{π}{2} \cdot \frac{(2 m - 1)!!}{(2 m)!!} = 2^{2 m} \frac{(2 m - 1)!!}{2^{m} m!} = \frac{(2 m)!}{m! m!}$ , where the last equality is $(2 m)! = 2^{m} m! (2 m - 1)!!$ of [F3].

4.1givenstep 1.1step 3.1∎

Conclusion: step 1.1 gives the vanishing for odd $k$ (including ${\tilde{p}}_{1} = 0$ by convention) and steps 1.2, 2.1, 3.1 give ${\tilde{p}}_{2 m} [Ω] = (\binom{2 m}{m})$ for every $m \geq 1$ .

Depends on

Used by

Scaled Plancherel profile moments converge in probability Proposition

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Sources

Vladimir Ivanov and Grigori Olshanski, Kerov's central limit theorem for the Plancherel measure on Young diagrams, arXiv:math/0304010; survey-paper version in Symmetric Functions 2001, NATO Science Series II 74 (2002), 93-151 (standard reference, not scraped)

The profile moments of ΩΩ are central binomial coefficients