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<p>We propose a new approach to conjugation-invariant random permutations. Namely, we explain how to construct uniform permutations in given conjugacy classes from certain point processes in the plane. This enables the use of geometric tools to study various statistics of such permutations. For their longest decreasing subsequences, we prove universality of the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2\sqrt{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <msqrt> <mi>n</mi> </msqrt> </mrow> </math></EquationSource> </InlineEquation> asymptotic. For Robinson–Schensted shapes, we prove universality of the Vershik–Kerov–Logan–Shepp limit shape, thus solving a conjecture of Kammoun. For the number of records, we establish a phase transition phenomenon as the number of fixed points grows. For pattern counts, we obtain an asymptotic normality result, partially answering a conjecture of Hamaker and Rhoades.</p>

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A geometric approach to conjugation-invariant random permutations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月8日

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  • Victor Dubach

摘要

We propose a new approach to conjugation-invariant random permutations. Namely, we explain how to construct uniform permutations in given conjugacy classes from certain point processes in the plane. This enables the use of geometric tools to study various statistics of such permutations. For their longest decreasing subsequences, we prove universality of the \(2\sqrt{n}\) 2 n asymptotic. For Robinson–Schensted shapes, we prove universality of the Vershik–Kerov–Logan–Shepp limit shape, thus solving a conjecture of Kammoun. For the number of records, we establish a phase transition phenomenon as the number of fixed points grows. For pattern counts, we obtain an asymptotic normality result, partially answering a conjecture of Hamaker and Rhoades.