Permuton and local limits for the Luce model

TL;DR

This work analyzes permutations drawn from the Luce model, where sampling without replacement is driven by prescribed positive weights. Under minimal weight assumptions, it establishes a permuton limit with an explicit density and proves convergence of pattern densities, along with a quenched Benjamini–Schramm local limit and a central limit theorem for consecutive pattern occurrences, plus a CLT for inversions. The results encompass generic weights and applications to Sukhatme and exponential Sukhatme weights, with precise formulas for limiting densities and variances. The findings illuminate global and local structural properties of Luce-permuted sequences and lay groundwork for statistical applications and further open questions in permutation limit theory and cycle structure.

Abstract

We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities proportional to prescribed positive weights. These permutations arise in applications such as ranking models, the Tsetlin library, and related Markov processes. Under minimal assumptions on the weights, we establish a permuton limit theorem describing the global behavior of Luce-distributed permutations and derive an explicit density of the limiting permuton. We further compute limiting pattern densities and analyze the differences between exact Luce permutations and their permuton approximations. We also study the local convergence of these permutations, proving a quenched Benjamini--Schramm limit and a central limit theorem for consecutive pattern occurrences. Finally, we prove a central limit theorem for the number of inversions.

Figures (6)

• Figure 1: From left to right: (1) The diagram of a Luce-distributed permutation with Sukhatme weights of size $10000$; (2) The density $\rho(x,y)$ of the permuton limit of Luce-distributed permutation with Sukhatme weights; (3) The diagram of a Mallows distributed permutation with parameter $q=(1-6/n)$ of size $n=10000$; (4) The density $\rho_6(x,y)$ of the permuton limit $\mu_6$ of Mallows distributed permutation with parameter $q=(1-6/n)$.
• Figure 2: Simulations for the proportion of consecutive patterns $\mathop{\mathrm{\widetilde{c\text{-}occ}}}\nolimits(\pi,\sigma_n)$ when $\sigma_n$ is a Luce-distributed permutation of size $n$ with Sukhatme weights $\theta_i=n-i+1$. In blue we show the histogram (renormalized to be a probability distribution) of the data collected from $3000$ random samples of size $1000$. In red we plot the density of $\mathcal{N}(\frac{1}{k!},\nu_\infty(\pi)/1000)$. From left to right: (1) The case of descents, i.e. $\pi=21$. (2) The case of $\pi=321$ (3) The case of $\pi=231$.
• Figure 3: Simulations for the proportion of consecutive patterns $\mathop{\mathrm{\widetilde{c\text{-}occ}}}\nolimits(\pi,\sigma_n)$ when $\pi\in\mathcal{S}_2$ and $\sigma_n$ is a Luce-distributed permutation of size $n$ with exponential Sukhatme weights $\theta_i=2^{n-i+1}$. In blue we show the histogram (renormalized to be a probability distribution) of the data collected from $3000$ random samples of size $1000$. In red we plot the density of $\mathcal{N}(\Lambda(\pi),\nu_\infty(\pi)/1000)$ with the values of $\Lambda(\pi)$ and $\nu_\infty(\pi)$ given in \ref{['eq:mean1']} and \ref{['eq:var1']}. From left to right: (1) The case of descents, i.e. $\pi=21$. (2) The case of $\pi=12$.
• Figure 4: Simulations for the proportion of consecutive patterns $\mathop{\mathrm{\widetilde{c\text{-}occ}}}\nolimits(\pi,\sigma_n)$ when $\pi\in\mathcal{S}_3$ and $\sigma_n$ is a Luce-distributed permutation of size $n$ with exponential Sukhatme weights $\theta_i=2^{n-i+1}$. In blue we show the histogram (renormalized to be a probability distribution) of the data collected from $3000$ random samples of size $1000$. In red we plot the density of $\mathcal{N}(\Lambda(\pi),\nu_\infty(\pi)/1000)$ with the values of $\Lambda(\pi)$ and $\nu_\infty(\pi)$ given in \ref{['eq:mean2']} and \ref{['eq:var2']}. From top-left to bottom-right: (1) $\pi=123$, (2) $\pi=132$, (3) $\pi=213$, (4) $\pi=312$, (5) $\pi=231$, (6) $\pi=321$.
• Figure 5: Simulations for the proportion of inversions $\widetilde{\mathop{\mathrm{occ}}\nolimits}(21,\sigma_n)=\frac{\mathop{\mathrm{occ}}\nolimits(21,\sigma_n)}{\binom{n}{2}}$ when $\sigma_n$ is a Luce-distributed permutation of size $n$ with Sukhatme weights $\theta_i=n-i+1$. In blue we show the histogram (renormalized to be a probability distribution) of the data collected from $3000$ random samples of size $n=1000$. In red we plot the density of $\mathcal{N}(1-\log(2),\frac{4 \times 0.0181166}{n})$.

Theorems & Definitions (20)

• Remark 1.1
• Theorem 2.1: Permuton limit
• proof : Proof of \ref{['thm:permuton_conv']}
• Proposition 2.2
• proof : Proof of \ref{['prop:density']}
• Corollary 2.3
• Proposition 2.4
• proof
• Theorem 3.1: Local limit
• Theorem 3.2