Classical patterns in Mallows permutations

Abstract

We study classical pattern counts in Mallows random permutations with parameters $(n,q_n)$, as $n\to\infty$. We focus on three different regimes for the parameter $q = q_n$. When $n^{3/2}(1-q)\to0$, we use coupling techniques to prove that pattern counts in Mallows random permutations satisfy a central limit theorem with the same asymptotic mean and variance as in uniformly random permutations. When $q\to1$ and $n(1-q)\to\infty$, we use results on the displacements of permutation points to find the order of magnitude of pattern counts. When $q\in(0,1)$ is fixed, we use the regenerative property of the Mallows distribution to compare pattern counts with certain $U$-statistics, and establish central limit theorems. We also construct a specific Mallows process, that is a coupling of Mallows distributions with $q$ ranging from $0$ to $1$, for which the process of pattern counts satisfies a functional central limit theorem.

Figures (4)

• Figure 1: The first figure shows a simulation of $\left( {\mathrm{inv}}\left(\tau_{n,t}\right) \right)_{t\in(0,q]}$ and its first-order limit $\left( n\, e_{2 1,t} = \frac{nt}{1-t} \right)_{t\in(0,q]}$. The other three show simulations of the centered process $\left( {\mathrm{inv}}\left(\tau_{n,t}\right) - n\, e_{2 1,t} \right)_{t\in(0,q]}$, approximating $\sqrt n \cdot \left( X_{2 1,t} \right)_{t\in(0,q]}$. This is done with $n=5000$ and up to time $q=0.8$. Note that the variance of the process $\left( X_{2 1,t} \right)$ seems to increase as $t$ gets closer to $1$: this is unsurprising when taking into account the fact that ${\mathrm{Occ}}\left(\pi,\tau_{n,t_n}\right)$ has a higher order of magnitude when $t_n \underset{n\to\infty}{\longrightarrow} 1$.
• Figure 2: The first two figures show simulations of $\left( {\mathrm{Occ}}\left(\pi,\tau_{n,t}\right) \right)_{t\in(0,q]}$ and $\left( {\mathrm{Occ}}\left(\pi,\tau_{n,t}\right) - \binom{n}{2} e_{\pi,t} \right)_{t\in(0,q]}$ for $\pi=213$. The other two show simulations of $\left( {\mathrm{Occ}}\left(\pi,\tau_{n,t}\right) \right)_{t\in(0,q]}$ for $\pi\in \{312, 3421\}$, for which the expression of $e_{\pi,t}$ is unknown. This is done with $n=5000$ and up to time $q=0.8$.
• Figure 3: Illustration of our coupling, and how it adapts to errors. Here, the first error happens at step $k=5$ (in orange). The following left-inversion counts are coupled using the functions $\phi_{\tau^5} : x \mapsto x + \mathbf{1}_{x<3}$ and $\phi_{u^5} : x \mapsto x + \mathbf{1}_{x<5}$. No error happens at step $6$, since $\phi_{\tau^5}(L_6) = 4 = \phi_{u^5}(U_6)$. Observe that ${\mathrm{pat}}\left([6]\setminus\{5\},\tau^6\right) = {\mathrm{pat}}\left([6]\setminus\{5\},u^6\right)$, as expected.
• Figure 4: On the left: the intervals defined in the proof of the upper bound of \ref{['th: order of occ for transition regime']}. Under the event $E$, all permutation points are contained in the grey area. On the right: the intervals defined in the proof of the lower bound of \ref{['th: order of occ for transition regime']}. The points with index in $I_k^{(2)}$, resp. $J_k^{(2)}$, are in the yellow area, resp. red area. With high probability, a positive proportion of points in the red area are in the yellow area.

Theorems & Definitions (55)

• Theorem 1.1: JNZ15
• Theorem 1.2
• Lemma 2.1
• Corollary 2.2
• Theorem 2.3
• Theorem 2.4
• Proposition 2.5
• Theorem 2.6
• Theorem 2.7
• Proposition 2.8