Asymptotic expansions relating to the lengths of longest monotone subsequences of involutions

TL;DR

The paper addresses the finite-size behavior of the length of the longest monotone subsequence in random involutions, establishing systematic asymptotic expansions in powers of $n^{-1/6}$ (general case) and $n^{-1/3}$ (fixed-point free case). The authors develop a generalized Poissonization/de-Poissonization framework tied to hard-edge/Laguerre ensembles and Baik–Rains/TW edge laws, proving that expansion terms are linear combinations of higher derivatives of the limiting distribution $F_β$ with rational coefficients, under a tameness hypothesis. They derive explicit first-order terms and provide extensive evidence (analytic, algebraic, and numerical) supporting the linear-form structure, along with kernel expansions, operator-determinant expansions, and Poissonization techniques to obtain uniform, multi-term asymptotics for distributions, discrete densities, and moments. The results significantly refine finite-size corrections for combinatorial and random-matrix models, strengthen connections between combinatorics of involutions and random matrix theory, and yield practical finite-n approximations validated by exact computations up to $n=1000$ and numerical experiments. Altogether, the work extends the toolkit for edge-universal phenomena in combinatorial probability and furnishes practical finite-size corrections for LIS-type statistics in involutions.

Abstract

We study the distribution of the length of longest monotone subsequences in random (fixed-point free) involutions of $n$ integers as $n$ grows large, establishing asymptotic expansions in powers of $n^{-1/6}$ in the general case and in powers of $n^{-1/3}$ in the fixed-point free cases. Whilst the limit laws were shown by Baik and Rains to be one of the Tracy-Widom distributions $F_β$ for $β=1$ or $β=4$, we find explicit analytic expressions of the first few expansion terms as linear combinations of higher order derivatives of $F_β$ with rational polynomial coefficients. Our derivation is based on a concept of generalized analytic de-Poissonization and is subject to the validity of certain hypotheses for which we provide compelling (computational) evidence. In a preparatory step expansions of the hard-to-soft edge transition laws of L$β$E are studied, which are lifted into expansions of the generalized Poissonized length distributions for large intensities. (This paper continues our work arXiv:2301.02022, which established similar results in the case of general permutations and $β=2$.)

Figures (5)

• Figure 1: Top row $\beta=1$; bottom row $\beta=4$. Plots of $E_{\beta,1}(t)$ (left panels) and $E_{\beta,2}(t)$ (middle panels) as in \ref{['eq:F_beta_j']}. The right panels show $E_{\beta,3}(t)$ as in \ref{['eq:F_beta_j']} (black solid line) with the approximation \ref{['eq:Fbeta3']} for $\nu=100$ (red dotted line) and $\nu=800$ (green dashed line): the close agreement validates the functional forms given in \ref{['eq:F_beta_j']}. Details about the numerical method can be found in MR2895091MR2600548arxiv.2206.09411MR3647807.
• Figure 2: Top row $\beta=1$; bottom row $\beta=4$. Plots of $F_{\beta,1}(t)$ (left panels) and $F_{\beta,2}(t)$ (middle panels) as in \ref{['eq:FbetaP']}. The right panels show $F_{\beta,3}(t)$ as in \ref{['eq:FbetaP']} (black solid line) with the approximation \ref{['eq:FbetaP3']} for $r=160$ (red dotted line) and $r=1200$ (green dashed line); the parameter $\nu$ has been varied such that $t_{\nu}(2r)$ covers the range of $t$ on display. Note that the functions $F_{\beta,j}(t)$ ($j=1,2,3$) are about two orders of magnitude smaller in scale than their counterparts in Fig. \ref{['fig:hard2soft']}.
• Figure 3: Top row $\varoast=\boxslash$; bottom row $\varoast=\boxbslash$. Plots of $F_{\varoast,1}(t)$ (left panels; both agree with the simulation-based approximation of their graphical form in arxiv.2205.05257) and $F_{\varoast,2}(t)$ (middle panels) as in \ref{['eq:FbetaD']}. The right panels show $F_{\varoast,3}(t)$ as displayed in \ref{['eq:FbetaD']} (black solid line) next to the approximations \ref{['eq:FbetaD3']} for $n=250$ (red $+$), $n=500$ (green $\circ$) and $n=1000$ (blue $\bullet$); the integer $l$ has been varied such that $t_{l^\varoast}(2n)$ covers the range of $t$ on display. The evaluation of \ref{['eq:FbetaD3']} uses a table of exact values of $p_\varoast(n;l)$ up to $n=1000$ (see Sect. \ref{['sect:exact']}).
• Figure 4: First three panels: plots of $F_{\boxdot,j}(t)$, $j=2,3,4$. Last panel: plot of $\sum_{j=5}^m F_{\boxdot,j}(t) \cdot n^{(5-j)/6}$ for $m=5$ (dotted red line), $m=6$ (dashed green line), $m=7$ (solid black line) vs. the right rand side of \ref{['eq:Fdot5']} for $n=1000$ (blue $\bullet$); the integer $l$ has been varied such that $t_{l+1}(n)$ covers the range of $t$ on display. The evaluation of \ref{['eq:Fdot5']} uses a table of exact values of the probabilities $p_\boxdot(n;l)$ for $n=1000$ (see Sect. \ref{['sect:exact']}). The choice of $m=7$ in \ref{['eq:Fdot5']} uses all the expressions displayed in \ref{['eq:Fdot']} and exhibits an excellent agreement (whereas $m=5$ is insufficient, reflecting that $n^{-1/6}\approx 0.316$ is a comparatively large quantity here).
• Figure 5: The exact discrete length probabilities for $n=1000$ (blue bars centered at the integers $l$) vs. their asymptotic expansions \ref{['eq:PDF_expan']} with $m=0$ (the Baik–Rains limit laws; dotted lines) and with $m$ chosen such that the error improves by a factor of $O(n^{-2/3})$ (solid line). The expansions are displayed as functions of a continuous variable $\nu$, evaluating the right-hand-side of \ref{['eq:PDF_expan']} with $\nu$ replacing the integer $l$. Left panel: $p^*_\boxslash(n;l)$ vs. the choice $m=2$ (solid line). Middle panel: $p^*_\boxbslash(n;l)$ vs. the choice $m=2$ (solid line). Right panel: $p^*_\boxdot(n;l)$ vs. the choice $m=5$ (solid line). The exact values are from the tables compiled in Sect. \ref{['sect:exact']}. Note that a graphically accurate continuous approximation of the discrete distribution must intersect the bars right in the middle of their top sides: this is the case for the choices of $m$ made for the solid lines. In contrast, the uncorrected limit laws (dotted lines) are noticeable inaccurate.

Theorems & Definitions (43)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.1
• proof
• Remark 2.4
• Theorem 2.2
• Corollary 2.1
• Theorem 3.1
• Remark 3.1