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added under “Properties” (here) brief statement of and pointer to the hook-content formula.
added this pointer:
Now I have put it all together and summed up (here) the full asymptotic expansion of the number of height-bounded standard Young tableaux. Currently I get this expression:
$\begin{aligned} & ln \big( \left\vert sYT_n(N) \right\vert \big) \\ & \; \underset{ { n \to \infty } \atop { N \to \infty } }{\sim}\; n ln(N) - \tfrac{1}{4}N^2 ln(n) + \tfrac{1}{4}N ln(n) \\ & \phantom{ \underset{ { n \to \infty } \atop { N \to \infty } }{\sim}\; } + \tfrac{1}{2} N^2 ln(N) - \left( \tfrac{3}{8} + \tfrac{ln(2)}{4} \right) N^2 - \tfrac{1}{4} N ln(N) + \tfrac{ln(2)}{2} N - \tfrac{1}{6} ln(N) \\ & \; \phantom{ \underset{ { n \to \infty } \atop { N \to \infty } }{\sim}\; } + \big( 2 \zeta^'(-1) - \tfrac{ln(2)}{6} - ln G(1/2) \big) \end{aligned}$I have checked factors a few times, but it won’t hurt to check again… (and thanks for catching a wrong factor in #6!)
Are there issues about taking limits in an order?
Next would be to get an approximation for min-entropy.
Are there issues about taking limits in an order?
I was wondering about that, too. Maybe this needs more thinking.
Incidentally, I was just turning again to Mkrtchyan 14. Its main Theorem 1.1. (p. 2) says something about the limit $n , N \to \infty$ taken with fixed proportionality, namely such that $n \propto N^2$.
In words, the statement of the theorem is roughly that in this limit the log of the Schur-Weyl probability density becomes “almost certainly approximately constant” , and that this constant scales as $\sqrt{n}$, hence as $N$ – which should mean that in this limit the Shannon entropy of the Schur-Weyl measure is approximately unity times this constant $\sim \sqrt{n} \cdot H_c$.
But the Shannon entropy of the Schur-Weyl measure is a lower bound for our entropy.
Not sure yet if this is good for anything, just saying.
In particular, this result (mentioned in #9) seems to imply that in the limit $n \propto N^2 \to \infty$ our Cayley-state probability distribution is
$(\lambda, i_\lambda) \;\mapsto\; \frac{ e^{\sqrt{n} \cdot H_c} }{dim\big(S^{(\lambda)}\big)} \;=\; \frac{ e^{\sqrt{n} \cdot H_c} }{ \left\vert sYTableaux_\lambda \right\vert } \,.$This would reduce us to computing asymptotic numbers of standard Young tableaux without bound on their height, which is a question with some classical results to it…
That’s interesting. There’s a sign out, no?
$(\lambda, i_\lambda) \;\mapsto\; \frac{ e^{-\sqrt{n} \cdot H_c} }{ \left\vert sYTableaux_\lambda \right\vert } \,.$So we have the hook content formula for $\left\vert sYTableaux_\lambda \right\vert$.
Moreover, if we assume that the Schur-Weyl- and Plancherel-measures are asymptotically indistinguishable (as suggested on p. 3 here, though the way in which the double scaling limit is taken may matter) then we could use in the above formula the asymptotic estimate
$ln \big( \left\vert sYTableaux_\lambda \right\vert \big) \;\sim\; \frac{c}{2}\sqrt{n} - \tfrac{1}{2}ln(n!)$(from here). Naively plugging in these asymptotic estimates into the formula for the Shannon entropy would then yield the following asymptotic expression for the entropy of the Cayley state:
$\begin{aligned} - \underset{ \mathclap{ {\lambda \in Part(n)} \atop { 1 \leq i_\lambda \leq \left\vert sYTableaux_\lambda \right\vert } } }{\sum} p^{Cay}(\lambda,i_\lambda) \cdot ln p^{Cay}(\lambda,i_\lambda) & = - \underset{ \mathclap{ {\lambda \in Part(n)} } }{\sum} p^{SW}(\lambda) \cdot ln \frac{e^{- H\cdot \sqrt{n}}}{ \left\vert sYoungTableaux_\lambda\right\vert } \\ & \underset{n = c N^2 \to \infty}{\sim} \underset{\lambda \in Part(n)}{\sum} e^{- \sqrt{n}\cdot H} \big( (- H - \tfrac{c}{2}) \cdot \sqrt{n} + \tfrac{1}{2} ln(n!) \big) \\ & \;\sim\; e^{\sqrt{n}\cdot (- H + \pi \sqrt{2/3} )} \big( (- H - \tfrac{c}{2}) \cdot \sqrt{n} + \tfrac{1}{2} ln(n!) \big) \end{aligned}$In the last step I used the Hardy-Ramanujan asymptotics for $\left\vert Part(n)\right\vert \sim \pi \sqrt{2n/3 }$ (taken from MO:103070 for the moment).
Not claiming that this is correct, as there is handwaving involved in putting these different asymptotics together this way. But maybe it helps to think along these lines.
Oh, I only now see #11. Yes, let me check the sign.
By the way, a joint scaling limit of just this form
$n = c N^2 \to \infty$underlies the AdS-CFT correspondence: there
$N = \lambda \left(\tfrac{1}{g}\right)^2 \to \infty$with $\lambda$ the ’t Hooft coupling.
Unfortunately I’ve got to spend today composing a reply to reviewers of a grant bid, with pretty poor prospects of success. This looks much more interesting.
Sure, no rush. And I need to get back to equivariant bundles.
BTW, I see now that the real asymptotic expansion of the partition function has a $1/n$ factor (now here).
With that included, the expansion of the Cayley entropy in #12 is of the form
$e^{ - a \cdot \sqrt{n} } \big( b + c \cdot n^{-1/2} + d \cdot n^{- 1/2} ln(n) \big) \,.$If here we use #14 to identify $n \propto g^{-2}$ and think of $g$ as a coupling constant, then this has the form of a non-perturbative contribution to a trans-series expansion of a Feynman series (here):
$e^{- \alpha /g} \big( \beta + \gamma \cdot g + \delta \cdot g ln(g) \big)$if we assume that the Schur-Weyl- and Plancherel-measures are asymptotically indistinguishable
I added a couple of papers to Pierre-Loïc Méliot which are relevant. Will need more time to digest.
At the bottom of page 2 of
there is discussion of the relationship between Schur-Weyl- and Plancherel-measures.
Could you check if you are pointing to the intended paper? On relation between the two measures this one seems to just mention that both have the same limit shape of Young diagrams for $c = 0$. (Incidentally, from p. 5 in Meliot 11 I take it that they don’t quite have the same limit shape for $c \neq 0$, but the statement there remains vague.)
I had been going by p. 3 of Mkrtchyan 11, which claims that at fixed $n$ the Schur-Weyl measure converges pointwise to the Plancherel measure as $N \to \infty$, with pointer, “for example”, to Olshanski 09 – I had dug that out but still can’t read it, since it is in Russian. But it would be necessary to see the details, since we would need such a convergence to hold not for fixed $n$, but for $n \propto N^2 \to \infty$.
On the other hand, for the argument in #12 to work we’d need less than the two measures to converge – It would be sufficient that they have the same null sets in the limit.
Sorry I got the wrong one. It’s bottom of p. 2 of
Thanks. So that’s indeed the kind of discussion as on that p. 5 in Meliot 11 that I had seen.
I suppose the fact that the limit shapes for $c \gt 0$ are different (though close, apparently) for the two measures means that the measures themselves can’t quite converge in the double scaling limit $\sqrt{n} = c N \to \infty$. Maybe they could still be close enough to have the same asymptotic null sets?
By the way, I was vaguely wondering if those limit shapes are the answer to our quest for the maximum of the Schur-Weyl distribution: We had seen that the naive idea that the maximum is at $\lambda = (n)$ receives some correction. But those limit shapes could possibly be thought of as a deformation of $\lambda = (n)$ by adding a small(er) amount of boxes in further rows. This intuition is supported by the graphics on the very top of p. 4 of the article you point to, for what it’s worth.
So I am wondering: Does the Schur-Weyl distribution in the limit $\sqrt{n} = c N \to \infty$ take its maximum on that “deformed LSKV” limit shape? Is that maybe implied by those “central limit theorems” that these authors discuss, saying that the Schur-Weyl measure is a Gaussian fluctuation around that limit shape, of sorts (if I am understanding that right)?
Two conditions given in the definition of a standard Young tableau seem to be non-equivalent. That is:
$\array{ 1 & 2 \\ 2 & 4 }$is a semistandard Young tableau in which rows and columns are strictly increasing and the last label is 4 (so the second condition is satisfied). But 3 does not appear in this table (so the first condition is not satisfied).
Thanks for the alert. This must be my bad. I remember thinking about how to best phrase this, and evidently I messed it up.
So I have changed the wording now to fix this; first saying simply that each label in $\{1, \cdots, n\}$ has to appear exactly once (here) and then remarking (here) on two implications of this constraint.
Just checked this didn’t carry over to Fundamental weight systems are quantum states, and I see we don’t actually define sYT when they appear briefly in Remark 3.21, just mention the fact that each label appears once.
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