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Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process

Anne-Laure Basdevant, Lucas Gerin, Maxime Marivain

TL;DR

The paper analyzes the length $L_n$ of the longest increasing subsequence for i.i.d. draws from a distribution with atoms. It introduces a variational framework with $f_t$ and $w_t$ and proves that $L_n$ is asymptotically governed by these quantities via $\displaystyle \frac{L_n}{\mathbb{E}[L_n]}\to1$ a.s., while $\limsup L_n/f_n\le1$ and $\liminf L_n/w_n\ge1$, revealing a broad spectrum of growth rates between constant and sub-$\sqrt{n}$ depending on tail behavior. The core method is a Poissonization and a coupling to an inhomogeneous Hammersley process, enabling non-asymptotic bounds and a variational characterization that unifies Poisson, geometric, and heavy-tailed cases, and extends to generic discrete distributions. The results show sharp asymptotics for several classical distributions and provide a pathway to understanding LIS phenomena in highly non-uniform discrete settings, with relevance to combinatorial probability and interacting particle systems. Overall, the work highlights how atom presence in the underlying distribution dramatically alters LIS growth and connects discrete LIS to a rich variational- and process-based framework.

Abstract

A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt{n})$, depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution.

Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process

TL;DR

The paper analyzes the length of the longest increasing subsequence for i.i.d. draws from a distribution with atoms. It introduces a variational framework with and and proves that is asymptotically governed by these quantities via a.s., while and , revealing a broad spectrum of growth rates between constant and sub- depending on tail behavior. The core method is a Poissonization and a coupling to an inhomogeneous Hammersley process, enabling non-asymptotic bounds and a variational characterization that unifies Poisson, geometric, and heavy-tailed cases, and extends to generic discrete distributions. The results show sharp asymptotics for several classical distributions and provide a pathway to understanding LIS phenomena in highly non-uniform discrete settings, with relevance to combinatorial probability and interacting particle systems. Overall, the work highlights how atom presence in the underlying distribution dramatically alters LIS growth and connects discrete LIS to a rich variational- and process-based framework.

Abstract

A famous result by Hammersley and Versik-Kerov states that the length of the longest increasing subsequence among iid continuous random variables grows like . We investigate here the asymptotic behavior of for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between and , depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of for a completely arbitrary distribution.

Paper Structure

This paper contains 17 sections, 17 theorems, 123 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Context and related results
  4. Description of the paper and strategy of proof
  5. Preliminaries: convergence of $L_n/\mathbb{E}[L_n]$
  6. Poissonization and coupling with an inhomogeneous Hammersley process
  7. Preliminaries for the proof of Theorem \ref{['th:poisson_domination']}: coupling with a Hammersley process
  8. Sources and sinks: stationarity
  9. Basic properties of $f_t$ and $w_t$
  10. Proof of Theorem \ref{['th:encadrement']}
  11. Proof of Theorem \ref{['th:encadrement']}: upper bound for $\mathbb{E}[L_n]$
  12. Proof of Theorem \ref{['th:encadrement']}: lower bound for $\mathbb{E}[L_n]$
  13. Asymptotics of $f_t$
  14. Preliminaries: a lower bound from a greedy algorithm
  15. Bounds for $f_t$
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $\mathbf{p}=(p_i)_{i\ge 1}$ be a discrete distribution on $\mathbb{Z}_{\geq 1}$, then Furthermore where $f_n,w_n$ are defined in eq:fneq:wn.

Figures (8)

  • Figure 1: A sample of $X_1,\dots,X_n$ for $n=200$ and a power law $p_i\asymp i^{-2.2}$. In red: a longest increasing subsequence of size $L_n=10$.
  • Figure 2: A sample of $X_1,\dots,X_n$ for $n=200$ and a geometric distribution $p_i\asymp 0.6^i$. In red: a longest increasing subsequence of size $L_n=9$.
  • Figure 3: A sample of $\Pi_t$ and of the corresponding process $(H(i))_{i\geq 1}$. Observe that $L\left( \Pi^{\leq i_0}_{t}\right)=4=\mathrm{card}(H(i_0))$, as stated by Proposition \ref{['prop:NombreLignes']}.
  • Figure 5: The same sample of $\Pi$ as \ref{['fig:schemasHammersley']}, with additional sources $\bullet$ and sinks $\bullet$ ($I^{(\alpha)}_2=I^{(\alpha)}_3=I^{(\alpha)}_5=1$ while other $I^{(\alpha)}_i$'s are equal to zero.) In blue: the corresponding process $(H^{(\alpha)}(i))_{i\geq 1}$. Proposition \ref{['prop:Stationnaire<']} states that (i) $H^{(\alpha)}(i_0)$ is also distributed as a homogeneous PPP with intensity $\alpha$ (ii) creations of particles on the right are distributed as sinks.
  • Figure 6: A simulation of the inhomogeneous Hammersley process $(H^{(\alpha)}(i))_{i\geq 0}$ with a distribution $p_i\asymp i^{-1.2}$, $\alpha=0.2$, sources and sinks distributed as in Proposition \ref{['prop:Stationnaire<']}, $n=100$. Time goes from bottom to top, points of $\Pi$ are depicted with $\bullet$'s. As in \ref{['fig:schemasHammersley_sources']}, sources/sinks/creations of particles are respectively depicted with $\bullet$/$\bullet$/$\bullet$. Proposition \ref{['prop:Stationnaire<']} states that: (i) Locations of particles at the top of the box are distributed as a homogeneous PPP with intensity $\alpha$. (ii) Spots $\bullet$ are distributed as $\bullet$.
  • ...and 3 more figures

Theorems & Definitions (41)

  • Remark 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Proposition 1.4: Asymptotics of $L_n$: Poisson and geometric distributions
  • Proposition 1.5: Asymptotics of $L_n$: heavy-tailed distributions
  • Theorem 1.6
  • Proposition 1.7
  • proof : Proof of Proposition \ref{['prop:variance']}
  • proof : Proof of \ref{['eq:Cv_ps_Theorem']}
  • Remark 2.1
  • ...and 31 more