Limit shape under Plancherel's law

[SlimKammoun]

Hello,

I think it would be nice to add a plot of the shape of the rescaled Young diagramme in Russian notations and the limit function $\Omega$.

I think that it would help to illustrate the Vershik-Kerov-Logan-Shepp limit shape theorem.

See for example https://arxiv.org/pdf/1212.3351.pdf page 32

[guilgautier]

That would be lovely indeed. Do you have time to think about it @SlimKammoun ? That could be useful to generate your figures to be used in articles :)

[SlimKammoun]

I will try :) ( I am not really in ease with graphics. But it's the time to learn :p)

[guilgautier]

Hey @SlimKammoun,

Would the following kind of display be satisfactory enough ?

I am planning to commit soon, so that you can play with it and give some feedback ;) Btw, to get the limit shape, I am actually rescaling by $\sqrt(theta)$ (see s.12) and not $theta$ (see p.32). Don't know if that's the right thing to do, I am a bit rusty ...

[SlimKammoun]

I obtained this graph diag.pdf in this commit : https://github.com/SlimKammoun/DPPy/commit/ff4d9439dc407fe54f24876fce7366e98e1fca23;

But yours is much more better thank you very much

[guilgautier]

None of them is better, they're the same 👍 The link to the commit doesn't work, can you update it please ?

Do you know what is the right rescaling to apply ?

Btw, to get the limit shape, I am actually rescaling by $\sqrt(theta)$ (see s.12) and not $theta$ (see p.32). Don't know if that's the right thing to do, I am a bit rusty ...

[SlimKammoun]

for the rescaling it depends on how to define the posonized Plancherel measure. The rescaling parameter is the average size of the Young diagram. So I think you have chosen the good scaling adequate to the definition you used before.

For the link of the commit (here

[guilgautier]

Hi @SlimKammoun, Can you play with the PoissonizedPlancherel object or simply visualize the output and give me some feedback with a comment or pull request ? If the current proposal suits you, you can close the issue 🤞

[SlimKammoun]

Great job @guilgautier. That's what I expected exactly. Thank you very much.