Side-by-side comparison of Plancherel growth process (Russian-notation Young diagram profile) and TASEP height function (step initial condition). Both produce piecewise-linear profiles with slopes plus or minus 1. Animated growth synchronized by box count, with limit shape overlays and accumulated histograms of height fluctuations at the center. Demonstrates that Plancherel pointwise bulk fluctuations are of order square-root log N, while TASEP fluctuations are O(N to the 1/6).
Plancherel and TASEP height functions look alike, but their fluctuations at the center differ dramatically: synchronized by box count $N$, Plancherel is $O(\sqrt{\log N})$ pointwise, while TASEP is $O(N^{1/6})$ (equivalently $O(t^{1/3})$).
Mathematical description
The Plancherel growth process adds boxes to a Young diagram via RSK insertion of i.i.d. uniform random variables. The resulting partition of $N$ is distributed according to the Plancherel measure. The TASEP (Totally Asymmetric Simple Exclusion Process) starts from step initial condition: particles at $\ldots, -2, -1, 0$, each jumping right at rate $1$ (i.e., $\mathrm{Exp}(1)$ waiting times), subject to the exclusion constraint.
Both processes produce piecewise-linear height functions with slopes $\pm 1$: the Russian-notation profile $\omega(u)$ for Plancherel, and $h(x,t)$ for TASEP ($+1$ over holes, $-1$ over particles). Both height functions have the shape of the Young diagram (a partition) sitting on top of the $|u|$ or $|x|$ baseline. The two processes are synchronized so that the total number of boxes (area of the partition) equals $N$.
Fluctuations at the center ($u = x = 0$): the Plancherel height $\omega(0) \approx \frac{4}{\pi}\sqrt{N}$ has $O(\sqrt{\log N})$ pointwise bulk fluctuations (Bogachev–Su), while the TASEP height $h(0,t) \approx t/2$ has $O(t^{1/3})$ KPZ fluctuations, i.e. $O(N^{1/6})$ under the box-count synchronization $N \asymp t^2$. Plancherel is dramatically more rigid. Regarding integrable structure: the Plancherel measure on partitions is a Schur measure, while the full TASEP height profile at fixed time is a diagonal marginal of an infinite Schur process on interlacing arrays: $x_i(t) \stackrel{d}{=} \lambda_i^{(i)} - i$. This coupling covers the entire curved rarefaction fan, both bulk and edge.
code
(note: parameters in the code might differ from the ones in simulation results below)-
Link to code(Interactive simulation — see source) -
Link to code(C++ source for WASM (RSK + TASEP algorithms))