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Local convergence for permutations and local limits for uniform \(\rho \)-avoiding permutations with \(|\rho |=3\)

Abstract

We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of consecutive pattern occurrences. We also characterize random limiting objects for this new topology introducing a notion of “shift-invariant” property (corresponding to the notion of unimodularity for random graphs). We then study two models in the framework of random pattern-avoiding permutations. We compute the local limits of uniform \(\rho \)-avoiding permutations, for \(|\rho |=3,\) when the size of the permutations tends to infinity. The core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern. For this result we use bijections between \(\rho \)-avoiding permutations and rooted ordered trees, local limit results for Galton–Watson trees, the Second moment method and singularity analysis.

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Acknowledgements

The author is very grateful to Mathilde Bouvel and Valentin Féray for introducing him to the fantastic world of random permutations. He also thanks them for the constant and stimulating discussions and suggestions. The author also warmly thanks Benedikt Stufler for precious suggestions about the terminology for Benjamini–Schramm convergence and for some explanations about local results for random trees. Finally, he thanks Tommaso Padovan for precious help in the realization of various simulations and the anonymous referee for all his/her precious and useful comments. This work was completed with the support of the SNF grant number \(200021\_172536\), “Several aspects of the study of non-uniform random permutations”.

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Appendices

Appeneix A: A generating function proof that \(P_{231}(\pi )\) is a probability distribution

We give here a generating function proof of the fact that

$$\begin{aligned} P_{231}(\pi ):=\frac{2^{|\text {LRMax}(\pi )|+|\text {RLMax}(\pi )|}}{2^{2|\pi |}},\quad \text {for all}\quad \pi \in \text {Av}(231), \end{aligned}$$
(74)

defines a probability distribution on \(\text {Av}^k(231),\) for all \(k\ge 1\).

Using our bijection between 231-avoiding permutations and binary trees, we can consider the generating function

$$\begin{aligned} C(z,x,y)=\sum _{T\in \mathcal {T}_b}z^{|T|}x^{|\text {LRMax}(\sigma _T)|}y^{|\text {RLMax}(\sigma _T)|}, \end{aligned}$$

where \(\mathcal {T}_b:=\mathbb {T}_b\cup \emptyset \) is the set of (possibly empty) binary trees. Using the classical decomposition of the binary trees in root, left subtree and right subtree, together with Observation 4.8, we obtain the following equation for our generating function,

$$\begin{aligned} C(z,x,y)=1+zxy\cdot C(z,x,1)\cdot C(z,1,y), \end{aligned}$$
(75)

where the term 1 corresponds to the empty tree.

Now setting \(x=y=1\) in Eq. (75) and \(A(z):=C(z,1,1),\) we obtain the classical relation for the generating function of Catalan numbers:

$$\begin{aligned}A(z)=1+z\cdot A(z)^2,\end{aligned}$$

that can be algebraically solved to yield \(A(z)=\frac{1-\sqrt{1-4z}}{2z}\).

Substituting \(y=1\) in Eq. (75) and setting \(B(z,x):=C(z,x,1)\), we have

$$\begin{aligned} B(z,x)= 1+ zx\cdot B(z,x)\cdot A(z) \end{aligned}$$

and so

$$\begin{aligned} B(z,x)=\frac{1}{1-zx\cdot A(z)}=\frac{2}{x\sqrt{1-4z}-x+2}. \end{aligned}$$

Finally, noting the symmetry \(C(z,x,1)=C(z,1,x)\), we obtain

$$\begin{aligned} \begin{aligned} C(z,x,y)&=1+zxy\bigg (\frac{2}{x\sqrt{1-4z}-x+2}\bigg )\bigg (\frac{2}{y\sqrt{1-4z}-y+2}\bigg )\\&=1+\frac{4zxy}{(2-x+x\sqrt{1-4z})(2-y+y\sqrt{1-4z})}. \end{aligned} \end{aligned}$$

Once we have the explicit expression for C(zxy),  in order to prove that Eq. (74) defines a probability distribution on \(\text {Av}^k(231)\), it suffices to notice that

$$\begin{aligned} C(z,2,2)=\frac{1}{1-4z}, \end{aligned}$$

and so \([z^n]C(z,2,2)=2^{2n}\).

Appendix B: A small lemma

Lemma B.1

Let \((\varvec{X}_n)_{n\in \mathbb {Z}_{>0}},\)\((\tilde{\varvec{X}}_n)_{n\in \mathbb {Z}_{>0}},\) be two sequences of real random variables with values in [0, 1] such that, for all \(n\in \mathbb {Z}_{>0},\)

$$\begin{aligned} \varvec{X}_n-\tilde{\varvec{X}}_n=\varvec{o}_P(1), \end{aligned}$$

i.e. \(\varvec{X}_n-\tilde{\varvec{X}}_n{\mathop {\rightarrow }\limits ^{P}}0\). Then, for every real continuous random process \(\varvec{Y}=(\varvec{Y}_t)_{t\in [0,1]}\) (i.e. a random variable with values in \(\mathcal {C}([0,1],\mathbb {R})\)) we have

$$\begin{aligned} \varvec{Y}_{\varvec{X}_n}-\varvec{Y}_{\tilde{\varvec{X}}_n}=\varvec{o}_P(1). \end{aligned}$$

Proof

Since \(\varvec{Y}=(\varvec{Y}_t)_{t\in [0,1]}\) is a.s. continuous on a compact set, it is also a.s. uniformly continuous. Therefore the continuity modulus \(\omega (\varvec{Y},\delta ):=\sup _{s,t\in [0,1],|s-t|\le \delta }|\varvec{Y}_t-\varvec{Y}_s|\) tends a.s. to zero as \(\delta \) tends to zero. By Skorokhod’s representation theorem we can assume that there exist a coupling such that \(\varvec{X}_n-\tilde{\varvec{X}}_n{\mathop {\rightarrow }\limits ^{a.s.}}0,\) therefore

$$\begin{aligned}|\varvec{Y}_{\varvec{X}_n}-\varvec{Y}_{\tilde{\varvec{X}}_n}|\le \omega (\varvec{Y},\varvec{X}_n-\tilde{\varvec{X}}_n){\mathop {\rightarrow }\limits ^{a.s.}}0.\end{aligned}$$

Hence we can conclude that \(\varvec{Y}_{\varvec{X}_n}-\varvec{Y}_{\tilde{\varvec{X}}_n}=\varvec{o}_P(1)\) in the original probability space. \(\square \)

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Borga, J. Local convergence for permutations and local limits for uniform \(\rho \)-avoiding permutations with \(|\rho |=3\). Probab. Theory Relat. Fields 176, 449–531 (2020). https://doi.org/10.1007/s00440-019-00922-4

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  • DOI: https://doi.org/10.1007/s00440-019-00922-4

Keywords

  • Local weak limits
  • Permutation patterns
  • Pattern-avoidance
  • Asymptotic distributions

Mathematics Subject Classification

  • 60C05
  • 05A05