A geometric approach to conjugation-invariant random permutations

TL;DR

This work introduces a geometric construction that realizes uniform permutations within a prescribed conjugacy class from planar point processes, enabling a unified treatment of conjugation-invariant random permutations. By translating permutation statistics into planar geometry, the authors establish universality results for longest monotone subsequences, the RS-limit shape, and the distribution of records and pattern counts, often without restrictive assumptions on cycle structure. Key technical ingredients include a planar PPP-based analysis, decomposition into locally uniform components, dependency graphs, and $U$-statistics, which together yield tight tails, limit curves $F_{LSKV}$, phase transitions in records, and CLTs for pattern counts. The framework both extends previous results for uniform permutations and resolves conjectures (notably Kammoun, Hamaker–Rhoades) under broad conditions, highlighting the robustness of geometric methods in random permutation statistics. The results have potential implications for representation-theoretic interpretations and for probabilistic combinatorics where conjugation-invariance plays a central role.

Abstract

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 $2\sqrt n$ asymptotic. For Robinson--Schensted shapes, we prove universality of the Vershik--Kerov--Logan--Shepp limit curve, 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.

Figures (6)

• Figure 1: The geometric construction of a random $t$-cyclic permutation for $t=(1,2,1,1)$. Here we used the arbitrary $t$-cyclic permutation $\mathfrak{s} := (1)\circ (2,3)\circ (4,5)\circ (6,7,8)\circ (9,10,11,12)$. The resulting permutation $\tau$ is written in one-line notation and in cycle product notation, and its cycles are highlighted on the point set. For example, the points $(U_6,U_7), (U_7,U_8), (U_8,U_6)$ are linked by the red dotted line, and correspond to the $3$-cycle $(9,4,12)$ in $\tau$.
• Figure 2: The dependency graph $\mathcal{L}_t$ associated with the cycle type $t=(2,1,2,0,1)$.
• Figure 3: Representation of the down-right sequence of cells (highlighted) encoded by the admissible sequence $((10,8),(8,7),(5,4),(4,4),(4,2))$, with $K=5$ and $\beta=2$.
• Figure 4: Left: a doubly increasing function $u\in\mathcal{U}_{0,r}\left( (0,1)^2 \right)$ for some $3\le r< 4$, with its integer level lines. Darker colors indicate greater values of $u$. The set $D(u)$ is highlighted in red. Right: in red, a $3$-decreasing point set $P$. The function $\kappa_P$ indeed satisfies $D(\kappa_P) = P$. Its values range from $0$ to $3$, with lighter colors indicating lower values and darker colors indicating higher values.
• Figure 5: The bands defined in the proof of \ref{['th: limit shape w/o fixed points']}. On each $A_i$ and each $B_i$, $\mathcal{P}$ is almost distributed like a homogeneous Poisson point process.

Theorems & Definitions (70)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• proof : Proof of \ref{['lem: geometric construction t-cyclic']}
• Theorem 2.1
• Corollary 2.2
• Proposition 2.3
• Corollary 2.4
• Conjecture 2.5
• Remark 1