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Asymptotic expansions relating to the distribution of the length of longest increasing subsequences

Folkmar Bornemann

Abstract

We study the distribution of the length of longest increasing subsequences in random permutations of $n$ integers as $n$ grows large and establish an asymptotic expansion in powers of $n^{-1/3}$. Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy-Widom distribution $F$, we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of $F$ with rational polynomial coefficients. Our proof replaces Johansson's de-Poissonization, which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is subject to a tameness hypothesis concerning complex zeros of the analytically continued Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft edge transition law of LUE is studied, which is lifted into an expansion of the Poissonized length distribution for large intensities. Finally, expansions of Stirling-type approximations and of the expected value and variance of the length distribution are given.

Asymptotic expansions relating to the distribution of the length of longest increasing subsequences

Abstract

We study the distribution of the length of longest increasing subsequences in random permutations of integers as grows large and establish an asymptotic expansion in powers of . Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy-Widom distribution , we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of with rational polynomial coefficients. Our proof replaces Johansson's de-Poissonization, which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is subject to a tameness hypothesis concerning complex zeros of the analytically continued Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft edge transition law of LUE is studied, which is lifted into an expansion of the Poissonized length distribution for large intensities. Finally, expansions of Stirling-type approximations and of the expected value and variance of the length distribution are given.
Paper Structure (26 sections, 17 theorems, 289 equations, 6 figures, 3 tables)

This paper contains 26 sections, 17 theorems, 289 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Expansions of perturbed Airy kernel determinants
  3. Bounds on the kernels and induced trace norms
  4. Fredholm determinants
  5. Expansions of operator determinants
  6. Expansion of the Hard-to-Soft Edge Transition
  7. Kernel expansions
  8. Proof of the general form of the expansion
  9. Functional form of $F_{1}(t)$, $F_{2}(t)$ and $F_3(t)$
  10. Functional form of $\tilde{F}_{1}(t)$ and $\tilde{F}_{2}(t)$
  11. Functional form of $\tilde{F}_{3}(t)$
  12. Lifting to the functional form of $F_{1}(t)$, $F_{2}(t)$, $F_{3}(t)$
  13. Simplifying the form of Choup's Edgeworth expansions
  14. Expansion of the Poissonized Length Distribution
  15. De-Poissonization and the Expansion of the Length Distribution
  16. ...and 11 more sections

Key Result

Lemma 1.1

Suppose the sequence $a_n$ of probabilities $0\leqslant a_n \leqslant 1$ satisfies the monotonicity condition $a_{n+1} \leqslant a_n$ for all $n=0,1,2,\ldots$ and denote its Poisson generating function by Then, for $s\geqslant 1$ and $n\geqslant 2\,$:Note the trade-off between sharper error terms $\mp n^{-s}$ and less sharp perturbations of $n$ by $\pm 2\sqrt{s n \log n}$.$P(n + 2\sqrt{s n \log n

Figures (6)

  • Figure 1: Plots of $F_{1}(t)$ (left panel) and $F_{2}(t)$ (middle panel) as in (\ref{['eq:F22']}a/b). The right panel shows $F_{3}(t)$ as in (\ref{['eq:F22']}c) (black solid line) with the approximations \ref{['eq:F23']} for $\nu=100$ (red dotted line) and $\nu=800$ (green dashed line): the close agreement validates the functional forms displayed in \ref{['eq:F22']}. Details about the numerical method can be found in MR2895091MR2600548arxiv.2206.09411MR3647807.
  • Figure 2: Plots of $F_{1}^P(t)$ (left panel) and $F_{2}^P(t)$ (middle panel) as in (\ref{['eq:F22P']}a/b). The right panel shows $F_3^P(t)$ as in (\ref{['eq:F22P']}c) (black solid line) with the approximations \ref{['eq:F23P']} for $r=250$ (red dotted line) and $r=2000$ (green dashed line); the parameter $\nu$ has been varied such that $t_\nu(r)$ covers the range of $t$ on display. Note that the functions $F_{j}^P(t)$ ($j=1,2,3$) are about two orders of magnitude smaller in scale than their counterparts in Fig. \ref{['fig:hard2soft']}.
  • Figure 3: Plots of $F_{1}^D(t)$ (left panel), $F_{2}^D(t)$ (middle panel) as in \ref{['eq:F22D']}; both agree with the numerical prediction of their graphical form given in the left panels of arxiv.2206.09411. The right panel shows $F_3^D(t)$ as in \ref{['eq:F22D']} (black solid line) with the approximations \ref{['eq:F23D']} for $n=250$ (red $+$), $n=500$ (green $\circ$) and $n=1000$ (blue $\bullet$); the integer $l$ has been varied such that $t_l(n)$ spreads over the range of $t$ displayed here. Evaluation of \ref{['eq:F23D']} uses the table of exact values of ${\mathbb P}(L_n\leqslant l)$ up to $n=1000$ that was compiled in arxiv.2206.09411.
  • Figure 4: Plots of $F_{1}^*(t)$ (left panel) and $F_{2}^*(t)$ (middle panel) as in (\ref{['eq:F22star']}a/b); both agree with the numerical prediction of their graphical form given in the right panels of arxiv.2206.09411. The right panel shows $F_3^*$ as in (\ref{['eq:F22star']}c) (black solid line) with the approximations \ref{['eq:F23star']} for $n=250$ (red $+$), $n=500$ (green $\circ$) and $n=1000$ (blue $\bullet$); the integer $l$ has been varied such that $t_{l-1/2}(n)$ spreads over the range of $t$ displayed here. Evaluation of \ref{['eq:F23star']} uses the table of exact values of ${\mathbb P}(L_n= l)$ up to $n=1000$ that was compiled in arxiv.2206.09411.
  • Figure 5: The exact discrete length distribution ${\mathbb P}(L_{n}=l)$ (blue bars centered at the integers $l$) vs. the asymptotic expansion \ref{['eq:PDFexpan']} for $m=0$ (the Baik--Deift--Johansson limit, dotted line) and for $m=2$ (the limit with the first two finite-size correction terms added, solid line). Left: $n=100$; right: $n=1000$. The expansions are displayed as functions of the continuous variable $\nu$, evaluating the right-hand-side of \ref{['eq:PDFexpan']} in $t=t_{\nu-1/2}(n)$. The exact values are from the table compiled in arxiv.2206.09411. 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, indeed, the case for $m=2$ (except at the left tail for $n=100$). In contrast, the uncorrected limit law ($m=0$) is noticeable inaccurate for this range of $n$.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Lemma 1.1: Johansson's de-Poissonization lemma MR1618351
  • Theorem 2.1
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Lemma 3.3
  • proof
  • ...and 28 more