# Asymptotic distribution of $\lambda_1$ under the $z$-measure for partitions

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.

(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.)

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.