# GOE Version of Longest Increasing Subsequence

Let $$S_n$$ be the symmetric group equipped with uniform measure. For any $$\pi\in S_n$$, let $$L_n=L_n(\pi)$$ denote the longest increasing subsequence. A celebrated result of Baik, Deift and Johansson states that

$$P\left(\frac{L_n-2\sqrt{n}}{n^{1/6}}\leq s\right)\rightarrow F_2(s),$$

where $$F_2(s)$$ is the Tracy-Widom distribution, describing the largest eigenvalue of a GUE matrix. The connection between longest increasing subsequence, representation theory and random matrix theory is now well established.

Question: is there another natural statistic on $$S_n$$ resembling something akin to $$L_n$$ which follows a GOE distribution? Specifically, calling such a statistic $$M_n$$, I'm interested in a similar result of the form

$$P\left(\frac{M_n-2\sqrt{n}}{n^{1/6}}\leq s\right)\rightarrow F_1(s),$$ where $$F_1(s)$$ is now the Tracy Widom distribution of the largest eigenvalue of a GOE. Ideally, the scaling $$n^{1/6}$$ should be the same. I would also prefer to retain uniform measure on $$S_n$$ but, of course I would be very interested to hear of other examples. As well, the shift can be different as well.

I'm aware of numerous results in combinatorics where $$F_1$$ comes up in. For example, the fluctations of random aztec diamond tilings follow an Airy process and therefore have GOE behavior. There the scaling is $$n^{1/3}$$. As well, it comes up in viscous walkers and nonintersecting brownian motion. I mention these examples because there are definitely tangentially linked to $$S_n$$ through representation theory and growth processes. However, as I said I would ideally like an example of something that naturally occurs on $$S_n$$.

Involutions $s=s^{-1}$ in $S_n$ are modeled by the Tracy-Widom distribution $F_1$ for real symmetric matrices (GOE): Take as $S_n^\ast$ the subset of involutions in $S_n$, and let $M_n$ be the corresponding random variable for the longest increasing subsequence. Then the limit distribution is $$P\left(\frac{M_n − 2\sqrt n}{n^{1/6}}\leq s\right)\rightarrow F_1(s).$$
The same paper also gives a generalization that is modeled by the Tracy-Widom distribution $F_4$ for the GSE (Gaussian symplectic ensemble). For a recent review article, see: