11
$\begingroup$

The following question about $z$-measures on Young diagrams came up in some ongoing work with Ofir Gorodetsky. I recall the background and then state our question below in the box.

For parameters $z$ and $z'$, define a measure on $\lambda \vdash n$ with the weights $$ M_{n,z,z'}(\lambda):= \frac{(\dim \lambda)^2}{n!\, (z z')_n} \prod_{\square \in \lambda} (z + c(\square))(z' + c(\square)). $$ Here $\dim \lambda$ is the number of standard Young tableaux of shape $\lambda$, the product is over all squares $\square$ in the diagram of $\lambda$, and $c(\square) := \mathrm{column}(\square) - \mathrm{row}(\square)$ is the content of $\square$. $(t)_n := t (t+1)\cdots (t + n-1)$. Definitions of the terminology used here can be found in e.g. vol. II of Stanley's Enumerative Combinatorics. Kerov discovered the remarkable fact that these weights induce a probability measure: $$ \sum_{\lambda \vdash n} M_{n, z, z'}(\lambda) = 1. $$ (At any rate they sum to $1$; they are a probability measure if all of them are non-negative.) Some of the proofs of this fact can be found for instance here or in section 8 here.

Let $\lambda^{(z,z',n)}$ be a random partition of $n$ according to these weights. We are interested in the limiting distribution of $\lambda_1$ under these weights in the case that $z = z' = \alpha > 0$. For $c \in [0,1]$, define $$ F_\alpha(c) := \lim_{n\rightarrow\infty}\mathbb{P}(\lambda_1^{(\alpha, \alpha, n)} \leq cn) = \lim_{n\rightarrow\infty}\sum_{\substack{\lambda \vdash n \\ \lambda_1 \leq cn}} M_{n,z,z'}(\lambda). $$ This limit is known to exist (and in fact converges to the distribution of $\alpha_1$ of the Thoma complex under the spectral z-measure; see this paper of Borodin). There are a variety of formulas characterizing the spectral z-measure in that paper (e.g. Theorem 2.2.1 for correlation functions of the Thoma point process), but we are interested in positivity of $F_\alpha(c)$ and have not been successful in applying these formulas to this. More exactly, we are interested in the following question:

Question: Is it true for each $\alpha \in (0,1)$, that $F_\alpha(c) > 0$ for all $c > 0$?

We are also interested in the behavior of $F_\alpha(c)$ when $\alpha > 1$ (we really understand matters completely only when $\alpha$ is a positive integer), and any information relevant would be interesting, but the question above is the one that is really pressing.

Below are some approximate graphs (generated from $M_{n,\alpha,\alpha}$ with $n=45$) of $F_\alpha(c)$ for $\alpha = .01, .1, .3, .5,$ and $.7$. The graph for $\alpha = .01$ is at the top and $\alpha = .7$ is on the bottom. $c$ of course is the horizontal axis.

$F_\alpha(c)$

(The apparent changes in convexity for $c$ near $1$ in the bottom two graphs are just a remnant of polynomial interpolation and should be ignored.)

$\endgroup$
4
$\begingroup$

A positive answer to Brad and mine's question is the main result of a recent paper of Olshanski, titled "The Topological Support of the z-Measures on the Thoma Simplex" (link). His proof is based on a previous work of his, as well as a recent paper of Korotkikh.

Interestingly, this question has (conjecturally) a simple number-theoretical interpretation, in terms of the distribution of the $z$th divisor function in short intervals and in arithmetic progressions. See Conjecture 4 in Brad and mine's preprint.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.