Long increasing subsequences in Brownian-type permutations

What is the behavior of the longest increasing subsequence of a uniformly random permutation? 
Its length is of order \(2n^{1/2}\) plus Tracy–Widom fluctuations of order \(n^{1/6}\). Its scaling limit is the directed geodesic of the directed landscape
 
This talk discusses how this behavior changes dramatically when one looks at universal Browniantype permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants \(1/2 < \alpha< \beta < 1\) such that the length of the longest increasing subsequence in a random permutation of size \(n\) sampled from the Brownian separable permutons is between \(n^{\alpha – o(1)}\) and \(n^{\beta + o(1)}\) with high probability. We present numerical simulations which suggest that the lower bound is close to optimal. 
Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). 
 
If time permits, we conclude by discussing some conjectures for permutations sampled from the skew Brownian permutons, a model of universal permutons generalizing the Brownian separable permutons: here, the longest increasing subsequences should be closely related with some models of random directed metrics on planar maps. 
 
Based on joint work with William Da Silva and Ewain Gwynne.
 
 

04/13/2023

Institute for Advanced Study, Princeton, New Jersey (USA), Probability seminar.

04/05/2023

The University of British Columbia, British Columbia (Canada), Probability seminar.

04/03/2023

The University of Washington, Washington state (USA), Probability seminar.

1,321 thoughts on “Long increasing subsequences in Brownian-type permutations”

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    最新のテクノロジーは、オンラインカジノとオンラインギャンブルの体験を一層豊かにしています。例えば、仮想現実(VR)技術の使用は、プレイヤーに没入型のギャンブル体験を提供し、実際のカジノにいるかのような感覚を生み出しています。また、ブロックチェーン技術の導入は、より透明で安全な取引を可能にし、プレイヤーの信頼を高めています。

    未来への展望
    オンラインカジノとオンラインギャンブルは、今後も技術の進歩とともに進化し続けるでしょう。人工知能(AI)の更なる統合、モバイル技術の発展、さらには新しいゲームの創造により、この分野は引き続き成長し、世界中のプレイヤーに新しい娯楽の形を提供し続けることでしょう。

    この記事では、オンラインカジノとオンラインギャンブルの現状、安全性、技術の影響、そして将来の展望に焦点を当てています。この分野は、技術革新によって絶えず変化し続ける魅力的な領域です。

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