Scaled Plancherel profile moments converge in probability

Monomial expansion: let $m={\tilde{p}}_{j_1}\cdots {\tilde{p}}_{j_i}$ be a monomial in the generators of $A$ of total weight $k=j_1+\cdots +j_i$, and fix $n\ge 1$; then $m$ lies in the weight filtration level $k$ of [F3], so its expansion $m={\sum }_{\rho }{m}_{\rho }{p}_{\rho }^{\mathrm{\#}}$ in the basis of [F3] has $m_{\rho }=0$ whenever $\mathrm{wt}(p_{\rho }^{\mathrm{\#}})>k$; and by [F2] the scaling identity applied to the $i$ factors gives $m(\bar{\lambda })=n^{-k\mathrm{/}2}m(\lambda (\cdot ))=n^{-k\mathrm{/}2}{\sum }_{\mathrm{wt}(\rho )\le k}{m}_{\rho }{p}_{\rho }^{\mathrm{\#}}(\lambda )$.

Pointwise integral identity: for every $k\ge 0$, every $\lambda ⊢n$ and $\omega \in D_0$ the definition of the profile moments in [F2] gives ${\tilde{p}}_{k+2}[\omega ]=\frac{(k+1)(k+2)}{2}{\int }_{\mathbb{R}}{x}^k(\omega (x)-\mathrm{\mid }x\mathrm{\mid })dx$; subtracting the same identity for $\omega =\mathrm{\Omega }$, which lies in $D_0$ by [F6], yields the pointwise identity ${\int }_{\mathbb{R}}(\bar{\lambda }(x)-\mathrm{\Omega }(x))x^{k}dx=\frac{2}{(k+1)(k+2)}({\tilde{p}}_{k+2}[\bar{\lambda }]-{\tilde{p}}_{k+2}[\mathrm{\Omega }])$ on $Y_n$.

Limit of expectations: by step 1.1 and [F1], ${\mathbb{E}}_{P_n}[m(\bar{\lambda })]=n^{-k\mathrm{/}2}{\sum }_{2r\le k}{m}_{(1^r)}{n}^{\downarrow r}$, and $n^{-k\mathrm{/}2}{n}^{\downarrow r}=n^{r-k\mathrm{/}2}{\prod }_{s=0}^{r-1}(1-s\mathrm{/}n)$ tends to $1$ when $2r=k$ and to $0$ when $2r<k$; hence ${\mathbb{E}}_{P_n}[m(\bar{\lambda })]$ tends to $m_{(1^{k\mathrm{/}2})}$ if $k$ is even and to $0$ if $k$ is odd. Define the linear functional $L$ on $A$ by $L(m):=m_{(1^{k\mathrm{/}2})}$ for a monomial of even weight $k$ and $L(m):=0$ for $k$ odd, extended linearly over the finitely many monomials of an element of $A$; then ${\mathbb{E}}_{P_n}[f(\bar{\lambda })]\to L(f)$ for every $f\in A$.

Multiplicativity: the functional $L$ of step 2.1 is the linear extension of the functionals $L_k$ of [F4], since for a monomial of weight $k$ the coefficient of $p_{(1^{k\mathrm{/}2})}^{\mathrm{\#}}$ is exactly the value of $L_k$ on its weight-$k$ component and lower-weight components contribute nothing; hence by the multiplicativity clause of [F4], applied to the weight-homogeneous components of two monomials, $L(m{m}^{\mathrm{\prime }})=L(m)L(m^{\mathrm{\prime }})$ for all monomials, and by linearity $L$ is multiplicative on $A$.

Evaluation at the limit profile: by [F4] one has $L({\tilde{p}}_{2m})=\left(\genfrac{}{}{0px}{}{2m}{m}\right)$ and $L({\tilde{p}}_k)=0$ for odd $k$, while by [F5] the evaluation functional $f\mapsto f[\mathrm{\Omega }]$ has exactly the same values on the generators ${\tilde{p}}_k$ and is multiplicative with $1[\mathrm{\Omega }]=1$; since $A$ is generated by the ${\tilde{p}}_k$, $L(f)=f[\mathrm{\Omega }]$ for every $f\in A$, and step 2.1 gives ${\mathbb{E}}_{P_n}[f(\bar{\lambda })]\to f[\mathrm{\Omega }]$ for every $f\in A$.

Convergence of the moments in probability: applying step 4.1 to $f={\tilde{p}}_j$ and to $f={\tilde{p}}_j^2$, and using that evaluation at $\mathrm{\Omega }$ is multiplicative, gives ${\mathbb{E}}_{P_n}[{\tilde{p}}_j[\bar{\lambda }]]\to {\tilde{p}}_j[\mathrm{\Omega }]$ and ${\mathbb{E}}_{P_n}[{\tilde{p}}_j[\bar{\lambda }{]}^2]\to {\tilde{p}}_j[\mathrm{\Omega }{]}^2$, so the variances of the random variables ${\tilde{p}}_j[\bar{\lambda }]$ on the finite probability space $(Y_n,P_n)$ tend to $0$; for every $\epsilon >0$, the mean differs from ${\tilde{p}}_j[\mathrm{\Omega }]$ by less than $\epsilon \mathrm{/}2$ for all sufficiently large $n$. The event $\mathrm{\mid }{\tilde{p}}_j[\bar{\lambda }]-{\tilde{p}}_j[\mathrm{\Omega }]\mathrm{\mid }\ge \epsilon$ is then contained in the event of deviation at least $\epsilon \mathrm{/}2$ from the current mean, so Chebyshev [F7] gives $\mathbb{P}(\mathrm{\mid }{\tilde{p}}_j[\bar{\lambda }]-{\tilde{p}}_j[\mathrm{\Omega }]\mathrm{\mid }\ge \epsilon )\le 4\mathrm{Var}({\tilde{p}}_j[\bar{\lambda }])\mathrm{/}{\epsilon }^2\to 0$, that is, ${\tilde{p}}_j[\bar{\lambda }]\to {\tilde{p}}_j[\mathrm{\Omega }]$ in probability for every $j\ge 2$.

Conclusion: for each fixed integer $k\ge 0$, put $j=k+2$; step 5.1 gives ${\tilde{p}}_j[\bar{\lambda }]\to {\tilde{p}}_j[\mathrm{\Omega }]$ in probability, and the identity of step 1.2 exhibits ${\int }_{\mathbb{R}}(\bar{\lambda }-\mathrm{\Omega })x^{k}dx$ as a fixed nonzero multiple of the difference, so ${\int }_{\mathbb{R}}(\bar{\lambda }-\mathrm{\Omega })x^{k}dx\to 0$ in probability as well; conversely the same identity transfers the latter convergence to the former. This proves the integral display and its equivalent moment form, and step 4.1 proves the assertion about every $f\in A$. No choice principle is used: all steps are finite computations on the finite probability spaces $(Y_n,P_n)$.