Expected size of a tree in the fixed point forest

TL;DR

This paper investigates the local limit of the fixed-point forest, an infinite random tree derived from a permutation sorting algorithm, and computes the expected size and leaves of its subtrees.

Contribution

It generalizes the fixed-point forest model and provides explicit calculations for the expected size, leaves, and variance bounds of subtrees within this model.

Findings

Expected size of a subtree is computed.

Expected number of leaves in a subtree is derived.

Bounds on the variance of subtree sizes are established.

Abstract

We study the local limit of the fixed-point forest, a tree structure associated to a simple sorting algorithm on permutations. This local limit can be viewed as an infinite random tree that can be constructed from a Poisson point process configuration on $[0,1]^\mathbb{N}$. We generalize this random tree, and compute the expected size and expected number of leaves of a random rooted subtree in the generalized version. We also obtain bounds on the variance of the size.

Figures

Equations

$$\pi^{(m)}(i)=\left\{\begin{array}[]{lr}m,&i=1\\ \pi(i-1),&2\leq i\leq m\\ \pi(i),&m<i\leq n\end{array}\right..$$

$$\mathbf{P}[G_n(r)=H]\to \mathbf{P}[G(r)=H].$$

$$\xi^{\prime}_{k}=\xi_{k+1}\Big{|}_{[0,x)}+\xi_{k}\Big{|}_{(x,1]}.$$

$$({\xi }_0^{{\pi }_n},\cdots ,{\xi }_{r-1}^{{\pi }_n}){⟶}_d({\xi }_0,\cdots ,{\xi }_{r-1})$$

$$\mathbf{E}[Y]=\mathbf{E}[\mathbf{E}[(Y|X)]]=1+\mathbf{E}[X]\mathbf{E}[Y]$$

$$\mathbf{E}[Y]=\frac{1}{1-\mathbf{E}[X]}.$$

$$\mathbf{E}[Y^2]=1+\mathbf{E}[X]\mathbf{E}[Y]+\mathbf{E}[X]\mathbf{E}[Y^2]+\mathbf{E}[X^2-X]\mathbf{E}[Y{]}^2,$$

$$\mathbf{E}[Y^2]=\frac{1}{(1-\mathbf{E}[X]{)}^2}+\frac{\mathbf{E}[X^2]-\mathbf{E}[X]}{(1-\mathbf{E}[X]{)}^3}.$$

$${\mathbf{P}}_{\alpha ,r}(|W|=n)=\frac{1}{n!}{e}^{-\alpha r}{\alpha }^n{r}^n$$

$${\mathbf{P}}_{\alpha ,r}(W=w)=\frac{1}{n!}{e}^{-\alpha r}{\alpha }^n.$$

$${\mathbf{P}}_{\alpha ,r}\left(\{W_A=u\}\cap \{|W|=n\}\right)=\frac{1}{n!}{e}^{-\alpha r}{\alpha }^n{r}^{n-j}.$$

$${\mathbf{P}}_{\alpha ,r}(W_A=u||W|=n)=r^{-j}$$

$$W_{a_i}=\#\{i<m\le j|{\sigma }_i^{-1}>{\sigma }_m^{-1}\}.$$

$$f_{y}(x)=\left\{\begin{array}[]{ll}x!,&x\leq y,\\ y!y^{x-y},&y<x.\end{array}\right.$$

$$|{\beta }_r(j)|=f_r(j)$$

$$|{\beta }_r(j)|=j!.$$

$${\mathbf{E}}_{\alpha ,r}[D^{(r)}]=\sum _{j=0}^r{\mathbf{E}}_{\alpha ,r}[D_j^{(r)}]$$

$${\mathbf{E}}_{\alpha ,r}[U^{(r)}]=\sum _{j=0}^{r-1}{\mathbf{E}}_{\alpha ,r}[U_j^{(r)}].$$

$${\mathbf{P}}_{\alpha ,r}\left(\{A\text{ is complete in }W\}\cap \{|W|=n\}\right)=e^{-\alpha r}{\alpha }^n{r}^{n-j}{f}_r(j)/n!.$$

$${\mathbf{E}}_{\alpha ,r}[D_j^{(r)}{1}_{|W|=n}]=\sum _{A\in \mathcal{A}}{e}^{-\alpha r}{\alpha }^n{r}^{n-j}{f}_r(j)/n!=e^{-\alpha r}{\alpha }^n{r}^{n-j}{f}_r(j)/(j!(n-j)!).$$

$${\mathbf{E}}_{\alpha ,r}[D_j^{(r)}{1}_{|W|=n}]=e^{-\alpha r}{\alpha }^n{r}^{n-j}/(n-j)!,$$

$${\mathbf{E}}_{\alpha ,r}[D_j^{(r)}]={\alpha }^j{e}^{-\alpha r}\sum _{n\ge j}\frac{(\alpha r{)}^{n-j}}{(n-j)!}={\alpha }^j.$$

$${\mathbf{E}}_{\alpha }[D]=\underset{r\to \infty }{\lim }{\mathbf{E}}_{\alpha ,r}[D^{(r)}]=\underset{r\to \infty }{\lim }\frac{1-{\alpha }^{r+1}}{1-\alpha }=\frac{1}{1-\alpha }.$$

$$\ell (r,n,A)=f_r(j)r^{{\sum }_{i=0}^{j-r}(a_{i+1}-a_i-1)}(r-1{)}^{{\sum }_{i=j-r+1}^j(a_{i+1}-a_i-1)}.$$

$$\ell (r,n,A)=j!(r-1{)}^{n-j}.$$

$$\mathbf{P}_{\alpha,r}\left(\Big{\{}|W|=n\Big{\}}\bigcap\Big{\{}X\text{ is a leaf}\Big{\}}\right)=e^{-\alpha r}\alpha^{n}(r-1)^{n-j}j!/n!.$$

$${\mathbf{E}}_{\alpha ,r}[U_j^{(r)}{1}_{\{|W|=n}\}]=\sum _{A\in \mathcal{A}}{e}^{-\alpha r}{\alpha }^n(r-1{)}^{n-j}j!/n!=e^{-\alpha r}{\alpha }^n(r-1{)}^{n-j}/(n-j)!.$$

$${\mathbf{E}}_{\alpha ,r}[U_j^{(r)}]=e^{-\alpha r}{\alpha }^j\sum _{n\ge j}(\alpha (r-1){)}^{n-j}/(n-j)!=e^{-\alpha }{\alpha }^j.$$

$${\mathbf{E}}_{\alpha }[U]=\underset{r\to \infty }{\lim }{\mathbf{E}}_{\alpha ,r}[U^{(r)}]=\underset{r\to \infty }{\lim }{e}^{-\alpha }\frac{1-{\alpha }^r}{1-\alpha }=\frac{e^{-\alpha }}{1-\alpha }.$$

$$x_r(A,B)=\frac{f_r(a+c)f_r(b+c)}{{\prod }_{a_i=b_j}\min (r,\max (a+c-i,b+c-j))}.$$

Full text

1 Introduction

We start with a simple sorting algorithm on a deck of cards labeled $1$ though $n$. If the value of the top card is $i$, place it in the $i$th position from the top in the deck. Repeat until the top card is a $1$. Viewing the deck of cards as a permutation in one-line notation $\pi=\pi(1)\pi(2)\cdots\pi(n)$, we create a new permutation, $\tau(\pi)$, by removing the value $\pi(1)$ from beginning of the permutation and putting it into position $\pi(1)$. For example, if $\pi=43512$ then $\tau(\pi)=35142$. This induces a graph whose vertices are the permutations of $[n]=\{1,\cdots,n\}$ and edges are pairs of permutations $(\pi,\tau(\pi)).$ Note that $\tau(\pi)$ has a fixed point at the position $\pi(1).$

This graph is a rooted forest, which we denote by $F_{n}$ and call the fixed point forest. A rooted forest is a union of rooted trees, and a tree is a graph that does not contain any closed loops involving distinct vertices. A permutation that begins with 1 is called the base of the tree in which they are contained. A thorough introduction to the fixed point forest can be found in Johnson et al. (2017).

The fixed point forest was first studied in McKinley (2015). The largest tree in $F_{n}$ has size bounded between $(n-1)!$ and $e(n-1)!$ and has as its base the identity permutation. The longest path from a leaf to a base is $2^{n-1}-1$ and is unique, starting from the permutation $23\cdots n1$ and ending at the identity.

Let $\mathfrak{S}_{n}$ denote the set of permutations of length $n$. For $\pi\in\mathfrak{S}_{n}$, let $\mathcal{F}(\pi)$ denote the collection of fixed points of $\pi$ other than $1$. For each $m\in\mathcal{F}(\pi)$ we create a new permutation $\pi^{(m)}$ such that

[TABLE]

We say we bump the value $m$ in $\pi$ to create $\pi^{(m)}$ and call $\pi^{(m)}$ a child of $\pi$. We let $\mathcal{C}(\pi)=\{\pi^{(m)}:m\in\mathcal{F}(\pi)\}$ denote the set of children of $\pi$. Every child $\sigma\in\mathcal{C}(\pi)$ satisfies $\tau(\sigma)=\pi$ hence is connected to $\pi$ in $F_{n}$.

Let $N(\pi)$ be the rooted tree in $F_{n}$ that contains $\pi$, with $\pi$ designated as the root instead of the unique permutation that starts with $1$ in $N(\pi)$. Let ${desc}(\pi)$ be the subtree of $N(\pi)$ rooted at $\pi$ and consisting of $\pi$ and its descendants, so that ${desc}(\pi)\subseteq N(\pi).$ We call this the descendant tree of $\pi$ (See Figure 1). Note that for any permutation $\sigma\in{desc}(\pi)$, there is some $r$ such that $\tau^{r}(\sigma)=\pi$.

By Theorem 3.5 in Johnson et al. (2017), there exists a tree, $\mathbf{T}$, such that as $n\to\infty$, for $\mathbf{\pi}_{n}$ chosen uniformly at random from permutations of size $n$, the randomly rooted tree $\mathbf{N}_{n}=N(\mathbf{\pi}_{n})$, converges in the local weak sense to $\mathbf{T}$. This limiting tree is described in Section 2 of Johnson et al. (2017), and the subtree of $\mathbf{T}$ which corresponds to the local weak limit of ${desc}(\mathbf{\pi}_{n})$ has a similar description, denoted by $\mathbf{D}$. In Johnson et al. (2017), they find the distribution for the shortest and longest paths from the root to a leaf in $\mathbf{D}$. The main purpose of the paper is to study the size of $\mathbf{D}$. For $\alpha\in[0,1]$, we define a generalization of $\mathbf{D}$, denoted $\mathbf{D}_{\alpha}$ such that $\mathbf{D}=\mathbf{D}_{1}$. We compute the expected size and expected number of leaves of $\mathbf{D}_{\alpha}$ and show that they are both unbounded for $\alpha=1$. Finally we find bounds on the second moment of the size of $\mathbf{D}_{\alpha}$. We show that the second moment has a phase transition from finite to infinite somewhere between $(3-\sqrt{5})/2$ and $(\sqrt{5}-1)/2.$

2 Local limits, point process configurations, and trees

Poisson Point Processes

The following briefly introduces an important probabilistic object: Poisson point processes. A thorough treatment can be found in Kingman (1993).

We say a random variable $X$ is $\mathrm{Poi}(\alpha)$ if it satisfies $\mathbf{P}(X=k)=\frac{1}{k!}e^{-\alpha}\alpha^{k}.$ If $X_{0}$ and $X_{1}$ are two independent $\mathrm{Poi}(\alpha_{0})$ and $\mathrm{Poi}(\alpha_{1})$, respectively, then their sum is $\mathrm{Poi}(\alpha_{0}+\alpha_{1})$.

A point process on $[0,1]$ is an integer-valued measure on Borel sets of $[0,1]$. It may be viewed as a collection of points, which represent the atoms of the measure. A point process configuration on $[0,1]$ is a collection of point processes, each on $[0,1]$, and can be viewed as a collection of labelled points on $[0,1].$

A Poisson point process on $[0,1]$ with intensity $\alpha$ is a random integer-valued measure which satisfies two properties: For any Borel subset $E\subset[0,1]$ with Borel measure $\lambda$, the number of atoms of the point process in $E$ is given by $\mathrm{Poi}(\alpha\lambda)$, and for any disjoint Borel subsets of $[0,1]$ the number of atoms in each are independent. Conditioned on the number of atoms in $E$ the location of each of the atoms is independent and uniform in $E$.

Collections of Poisson point processes can be merged to create a single poisson point process. Suppose $\xi_{0}$ is a $\mathrm{Poi}(\alpha_{0})$ point process on $[0,1]$ and $\xi_{1}$ is $\mathrm{Poi}(\alpha_{1})$ point process on $[0,1]$ with $\xi_{0}$ and $\xi_{1}$ both independent. Then the union of $\xi_{0}$ and $\xi_{1}$ is distributed like a $\mathrm{Poi}(\alpha_{0}+\alpha_{1})$ point process. The reverse is also true. Let $\xi^{\prime}$ be a $\mathrm{Poi}(\alpha_{0}+\alpha_{1})$ point process on $[0,1]$ and label each atom [math] with probability $\alpha_{0}/(\alpha_{0}+\alpha_{1})$ and $1$ otherwise. Let $\xi_{0}$ denote the point process consisting of the atoms labeled [math] and $\xi_{1}$ the point process of the remaining atoms. Then $\xi_{0}$ and $\xi_{1}$ are, respectively, independent Poisson($\alpha_{0}$) and Poisson($\alpha_{1}$) point processes on $[0,1]$. This can be generalized further to $\alpha=\alpha_{0}+\cdots+\alpha_{k-1}$. If $\xi^{\prime}$ is a Poisson($\alpha$) point process each atom in $\xi^{\prime}$ is independently labeled such that the label is $i$ with probability $\alpha_{i}/\alpha$ for $0\leq i<k$, then the collection of atoms labeled $i$ is a Poisson($\alpha_{i}$) point process and each $\xi_{i}$ is independent of the rest.

Let $\xi_{1}$ and $\xi_{2}$ be two independent Poisson($\alpha)$ point processes. For $x\in(0,1)$, define \xi^{\prime}_{1}=\xi_{2}\big{|}_{[0,x)}+\xi_{1}\big{|}_{(x,1]} to be the point process consisting of the atoms from $\xi_{2}$ restricted to the interval $[0,x)$ and the atoms from $\xi_{1}$ restricted to the interval $(x,1]$. If $x$ is independent of $\xi_{1}$ and $\xi_{2}$ then the resulting process $\xi^{\prime}_{1}$ is also a Poisson($\alpha$) point process.

Weak Convergence

We give a brief definition of the version of local weak convergence that is used to define $\mathbf{T}$ and $\mathbf{D}$. See Aldous and Steele (2004) or Benjamini and Schramm (2001) for a proper discussion of local weak convergence, which is sometimes referred to as Benjamini-Schramm convergence.

Let $G_{1},G_{2}\cdots$ be a sequence of rooted graphs. For any rooted graph $H$, the $r$-neighborhood of the root, denoted $H(r)$, is the subgraph of $H$ induced from all vertices that are distance at most $r$ from the root. The rooted graph $G$ is the local weak limit of $G_{n}$ if for every $r\geq 0$ and every finite graph $H$,

[TABLE]

From point process configurations to trees

Let $\xi=(\xi_{k})_{k\geq 0}$ be a point process configuration on $[0,1]^{\mathbb{N}}$ where each $\xi_{k}$ is a point process on $[0,1].$ For each atom $x\in\xi_{0}$ define the bump map $f(\xi,x)=(\xi^{\prime}_{k})_{k\geq 0}$ where

[TABLE]

See Figure 2 for an illustration of this map. Given a point process configuration, $\xi$, the bump map allows us to recursively define a tree with root $v_{0}$ whose vertices are point process configurations. Define $v_{0}$ to be the root of the tree with corresponding point process configuration $\xi^{v_{0}}=\xi$. Suppose $v$ is a vertex in the tree with corresponding point process configuration given by $\xi^{v}$. For each $x\in\xi^{v}_{0}$, create a new vertex $v(x)$ in the tree with point process configuration given by the bump map $\xi^{v(x)}=f(\xi^{v},x)$. The newly created vertex $v(x)$ is a considered a child of $v$. We call this tree the bump tree of $\xi$ and denote it by $\gamma(\xi).$ For fixed $r\geq 0$ let $\gamma_{r}(\xi)$ denote the $r$-neighborhood of the root in $\gamma(\xi).$ Only the atoms in $(\xi_{0},\cdots,\xi_{r-1})$ are necessary to determine the structure of the $\gamma_{r}(\xi)$, so we may write $\gamma_{r}(\xi)=\gamma_{r}(\xi_{0},\cdots,\xi_{r-1})$ and assume $\xi_{k}=\emptyset$ for $k\geq r$. The map $\gamma_{r}$ is continuous because a slight perturbation of the atoms will not change the relative order of the points in $(\xi_{0},\cdots,\xi_{r})$. See Figure 3 for an example of a finite neighborhood of the root of the bump tree for a point process configuration.

For a permutation $\pi$ of length $n$, we say the index $i$ or the value $\pi(i)$ is $k$-separated if $\pi(i)=i+k.$ We define the separation word of $\pi$ point-wise by $\mathbf{W}^{\pi}(i):=\pi(i)-i$. No two permutations have the same separation word. From this word we can construct a point process configuration $(\xi^{\pi}_{k})_{k\geq 0}$ by placing an atom in $\xi^{\pi}_{k}$ at position $i/n$ if $i$ is a $k$-separated point in $\pi$.

By Proposition 3.4 in Johnson et al. (2017), for fixed $r\geq 0$, as $n$ tends to infinity,

[TABLE]

where $\xi_{k}$ is a $\mathrm{Poi}(1)$ point process on $[0,1]$. From the arguments of Theorem 3.5 in Johnson et al. (2017), letting $\xi=(\xi_{k})_{k\geq 0}$, we have $\gamma_{r}(\xi^{\pi_{n}})\to\gamma_{r}(\xi)$ by continuity of $\gamma_{r}$ and the Continuous Mapping Theorem [Billingsley (1999)]. Furthermore, it is seen that $\gamma_{r}(\xi^{\pi_{n}})$ is the same as the $r$-neighborhood of the descendant tree ${desc}(\mathbf{\pi}_{n})$ with high probability. Therefore $\mathbf{D}:=\gamma(\xi)$ is the local weak limit of ${desc}(\mathbf{\pi}_{n})$.

We now can state our main results. For $\alpha\in(0,1]$, let $\xi=(\xi_{k})_{k\geq 0}$ be a collection of independent $\mathrm{Poi}(\alpha)$ point processes on $[0,1]$ and let $\mathbf{D}_{\alpha}:=\gamma(\xi)$ be the corresponding bump tree of $\xi$. Let $D$ denote the number of vertices and $U$ the number of leaves in $\mathbf{D}_{\alpha}.$ Finally let $\mathbf{E}_{\alpha}$ and $\mathbf{P}_{\alpha}$ denote the expectation and probability associated with $\mathrm{Poi}(\alpha)$ point processes. We now may state our main results.

Theorem 1**.**

For $0<\alpha<1$, $\mathbf{E}_{\alpha}[D]=(1-\alpha)^{-1}$, and $\mathbf{E}_{1}[D]$ diverges.

Theorem 2**.**

For $0<\alpha<1$, $\mathbf{E}_{\alpha}[U]=e^{-\alpha}(1-\alpha)^{-1}$, and $\mathbf{E}_{1}[U]$ diverges.

Theorem 3**.**

For $\alpha\geq(\sqrt{5}-1)/2$, $\mathbf{E}_{\alpha}(D^{2})$ diverges. For $\alpha<(3-\sqrt{5})/2$, $\mathbf{E}_{\alpha}(D^{2})$ is finite.

3 Comparison with Galton-Watson trees

In this section we compare our results to the well-studied Galton-Watson tree Watson and Galton (1875); Neveu (1986).

A Galton-Watson tree, $\mathbf{GW}$, can be constructed through a simple random process. Start with a root $v_{0}$ and a nonnegative integer-valued random variable $X$. Create $X_{v_{0}}$ children of $v_{0}$ where $X_{v_{0}}$ is distributed as and independent copy of $X$. For each child, $v$, of $v_{0}$ repeat this process, where $X_{v}$ is an independent copy of $X$. Depending on the distribution of $X$, the resulting tree will have drastically different behavior.

Fix a nonnegative integer-valued random variable $X$ with finite expectation $0<\mathbf{E}[X]<1$ and finite second moment $\mathbf{E}[X^{2}]<\infty.$ Let $Y=|\mathbf{GW}|$. Let $X$ denote the number of children of the root of $\mathbf{GW}$ and for $1\leq i\leq X$, let $Y^{i}$ denote the number of vertices in the subtree consisting of the $i$th child and all of its descendants. Each $Y^{i}$ is distributed identically as an independent copy of $\mathbf{GW}$. We denote the size of $\mathbf{GW}$ conditioned on $X$ by $(Y|X)=1+\sum_{i=1}^{X}Y^{i}$. Taking expectation we have $\mathbf{E}[(Y|X)]=1+X\mathbf{E}[Y]$ and thus

[TABLE]

and so

[TABLE]

A similar approach for the second moment gives the equation

[TABLE]

which can be simplified to

[TABLE]

Given that $\mathbf{E}[X]<1$ and $\mathbf{E}[X^{2}]$ is finite, (1) shows that $\mathbf{E}[Y^{2}]$ finite. In particular if $X$ is $\mathrm{Poi}(\alpha)$ then $\mathbf{E}[Y]$ agrees with $\mathbf{E}_{\alpha}[D]$ from Theorem 1, while Theorem 3 shows the second moment $\mathbf{E}[Y^{2}]$ cannot agree with the second moment $\mathbf{E}_{\alpha}[D^{2}]$ if $\alpha\geq(\sqrt{5}-1)/2$ since the former is finite while the latter diverges.

The approach used to compute $\mathbf{E}[Y]$ and $\mathbf{E}[Y^{2}]$ cannot be used to compute $\mathbf{E}_{\alpha}[D]$ and $\mathbf{E}_{\alpha}[D^{2}]$ because the subtrees from the root in $\mathbf{D}_{\alpha}$ are not independent of each other.

4 Words from point process configurations

For a collection of point processes on $[0,1]$, $\xi=\{\xi_{k}\}_{k\geq 0}$, let ${w}_{r}(\xi)$ be the word constructed from the relative order of the atoms in $(\xi_{0},\cdots,\xi_{r-1})$. For example see Figure 4. Assuming that no two atoms of $\xi$ are in the same location, the structure of the $r$-neighborhood of the root in the tree $\gamma_{r}(\xi)$ can be constructed directly from this word. Let $\Omega_{r}$ denote the space of finite words with letters from $\{0,\cdots,r-1\}$.

If $\xi$ is a $\mathrm{Poi}(\alpha)$ point process configuration, this induces a probability measure $\mathbf{P}_{\alpha,r}$ on $\Omega_{r}$ for every $r\geq 0$. The following lemma describes this distribution.

Lemma 4**.**

Let $\xi$ be a $\mathrm{Poi}(\alpha)$ point process configuration and $W={w}_{r}(\xi)$ the word given by the relative order of the first $r$ point processes of $\xi$. Let $w$ denote a fixed word of length $n$ in $\Omega_{r}$. Then

[TABLE]

and

[TABLE]

Proof.

Construct the $r$ independent $\mathrm{Poi}(\alpha)$ point processes from a single $\mathrm{Poi}(r\alpha)$ point process by labeling each atom independently from $\{0,\cdots,r-1\}$, choosing the label uniformly at random. The probability that $|W|=n$ is precisely the probability that a $\mathrm{Poi}(r\alpha)$ point process has $n$ atoms in $[0,1]$, the right hand side of (2). As the labeling is independent for each atom, each of the $r^{n}$ possible labelings is equally likely, so the probability that $W=w$ for a fixed $w$ of length $n$ is computed by dividing the right hand side of (2) by $r^{n}$, giving (3). ∎

For $W\in\Omega_{r}$ of length $n$ we write $W=W_{1}\cdots W_{n}$ in one line notation. For a fixed subset of indices $A=(i_{1},\cdots,i_{j})$ let $W_{A}=W_{i_{1}}\cdots W_{i_{j}}$. We may refine Lemma 4 even further.

Lemma 5**.**

Let $u=u_{1}\cdots u_{j}$ be a word in $\Omega_{r}$. Let $W\in\Omega_{r}$, and $A=(i_{1},\cdots,i_{j})$ be a set of indices such that $1\leq i_{1}<\cdots<i_{j}\leq n$. Then,

[TABLE]

Proof.

Conditioned on $|W|=n$, the labels of the atoms indexed by $A$ are chosen independently so

[TABLE]

and the statement follows. ∎

The tree $\gamma_{r}(\xi)$ with word ${w}_{r}(\xi)$ will agree up to a relabeling of the vertices of the tree $\gamma_{r}(\xi^{\prime})$ if ${w}_{r}(\xi)={w}_{r}(\xi^{\prime}).$ A vertex in the tree corresponds to bumping a particular set of atoms in a particular order. Therefore the measure $\mathbf{P}_{\alpha,r}$ on words in $\Omega_{r}$ is exactly the measure we need to understand the $\gamma_{r}(\xi)$.

We can translate our language of bumping atoms in $\xi$ to bumping letters in words. Let $W\in\Omega_{r}$. For each $0\in W$, we construct a new word by removing the chosen [math] and reducing every letter to the left of it by $1.$ We say the index of this letter [math] is bumped and indices less than the bumped index are shifted. The set of indices of the [math]s in a word are called the bumpable indices. The set of words that can be constructed by bumping a single [math] in $W$ are called the children of $W$ and denoted $\mathcal{C}(W).$ For example the word $2\ 1\ 0\ 1\ 0$ has has two children, $1\ 0\ \square\ 1\ 0$ and $1\ 0\ \square\ 0\ \square$, where $\square$ is used to indicate bumped indices or indices shifted below zero. Once the letter at an index becomes $\square$ in a word it can never become [math] in one of its descendants. We construct a rooted tree, denoted $\gamma(W)$, following a process that mirrors our construction of $\gamma(\xi)$ for point process configurations. We let $\gamma_{j}(W)$ denote the $j$-neighborhood of the root in $\gamma(W)$.

We may omit the $\square$ symbol in the labeling of the tree. The $\square$ symbol is used to emphasize that the set of indices is the same for each word in the same tree. See Figure 5 for the rooted tree in $\Omega_{3}$ associated with the word $2\ 1\ 0\ 1\ 0$. The sequence of indices that are bumped to reach the vertex $v$ in $\gamma(W)$ is called the bumping sequence of $v$.

For $j\geq 1$ and every vertex $v\in\gamma_{j}(W)\backslash\gamma_{j-1}(W)$ there is a corresponding set of $j$ atoms that must be bumped in a particular order to reach $v$. This sequence of atoms induces an ordered set of indices $A=\{a_{1}<\cdots<a_{j}\}$ and permutation, $\sigma$, of length $j$ such that $v$ is obtained by bumping the atoms at the indices in order $\{a_{\sigma_{1}},\cdots,a_{\sigma_{j}}\}$ where each of the indices must be [math] when they are bumped. We say the set of indices $A$ reaches $v$ by the order $\sigma$. Since $\gamma(W)$ is a tree, any such $v$ is reachable by a unique pair $(A,\sigma)$.

For a set of indices $A=\{a_{1}<\cdots<a_{j}\}$, we say $A$ is complete in $W$ if there exists an order $\sigma\in\mathfrak{S}_{j}$ and a sequence of words $W=W^{0},\cdots,W^{j}$ such that for $1\leq i\leq j$, $W^{i}\in\mathcal{C}(W^{i-1})$ is obtained by bumping the index $a_{\sigma_{i}}$ in $W^{i}$. Whether or not $A$ is complete in $W$ is independent of the letters not in $A$. The following lemma gives conditions on when $A$ is complete in $W$.

Lemma 6**.**

If $A$ is complete in $W\in\Omega_{r}$ with $|A|=j$, there is a unique $\sigma\in\mathfrak{S}_{j}$ such that a vertex in $\gamma(W)$ is reachable by $(A,\sigma)$. If $r\geq j$, then for each $\sigma\in\mathfrak{S}_{j}$ there is a unique sequence of values $u=u_{1}\cdots u_{j}$ such if $W_{A}=u$ then there exists a vertex in $\gamma(W)$ that is reachable by $(A,\sigma)$.

Finally, $A$ is complete with respect to $W$ if and only if $W_{a_{i}}\leq\min(j-i,r-1)$ for $1\leq i\leq j$.

Proof.

Since $A$ is complete in $W$ there is at least one $\sigma\in S_{j}$ and $v$ in $\gamma(W)$ such that $v$ is reachable by $(A,\sigma)$. First $a_{\sigma_{1}}$ is bumpable if and only if $W_{a_{\sigma_{1}}}=0.$ In order for $a_{\sigma_{i+1}}$ to be bumpable after bumping $a_{\sigma_{1}}$ up to $a_{\sigma_{i}}$, the label of $a_{\sigma_{i+1}}$ must be [math], and therefore index must be shifted exactly $W_{a_{\sigma_{i+1}}}$ times by bumping indices larger then $a_{\sigma_{i+1}}.$ For this to occur there must be exactly $W_{a_{\sigma_{i+1}}}$ integers $m$ such that $m<i+1$ and $\sigma_{m}>\sigma_{i+1}.$ In terms of $\sigma^{-1}$ we have for $1\leq i\leq j$,

[TABLE]

The sequence of values $W_{a_{1}}\cdots W_{a_{j}}$ is the unique inversion table (Knuth (1998)) for the permutation $\sigma^{-1}$. No two permutations have the same inversion table and thus $\sigma$ must be unique. Given a $\sigma\in\mathfrak{S}_{j}$, if $W_{A}$ is the inversion table for $\sigma^{-1}$ then $A$ will be complete with respect to $W$.

Finally we have that $W_{a_{1}}\cdots W_{a_{j}}$ is an inversion table if and only if $W_{a_{i}}\leq j-i$ for $1\leq i\leq j$. We also have that $W_{a_{i}}\leq r-1$ by definition. ∎

Define the following truncated factorial function:

[TABLE]

Note that $\lim_{y\to\infty}f_{y}(x)=x!$.

Let $\beta_{r}(j)$ denote the set of subwords of length $j$ such such that $A$ is complete in $W$ if and only if $W_{A}\in\beta_{r}(j)$. For any $r\geq 0$ and $j\geq 0$, by Lemma 6,

[TABLE]

and for $r\geq j$, this simplifies to

[TABLE]

5 Expectation of $D$ and $U$

Let $D^{(r)}$ denote the number of vertices in $\gamma_{r}(\xi)$. Let $U^{(r)}$ denote the number of leaves in $\gamma_{r}(\xi)$ that are distance less than $r$ from the root. Note that a leaf in $\gamma_{r}(\xi)$ that is distance $r$ from the root may not be a leaf in $\gamma_{r+1}(\xi).$ By Theorem 5.1 in Johnson et al. (2017), the longest path to a leaf in $\gamma(\xi)$ is almost surely finite and therefore $\gamma_{r}(\xi)$ is identical to $\gamma(\xi)$ for large enough $r.$ To compute the expectation of $D$ and $U$ it suffices to compute the expectation of $D^{(r)}$ and $U^{(r)}$ and let $r$ tend to infinity.

Let $W$ be chosen from $\Omega_{r}$. For $j\leq r$ let $D^{(r)}_{j}=|\gamma_{j}(W)\backslash\gamma_{j-1}(W)|.$ Similarly let $\mathcal{L}_{j}$ denote the set of leaves in $\gamma_{j}(W)$, so that for $j\leq r-1$, $U^{(r)}_{j}=|\mathcal{L}_{j}(W)\backslash\mathcal{L}_{j-1}(W)|$, the number of leaves in $\gamma_{j}(W)$ exactly distance $j$ from the root. By linearity of expectation

[TABLE]

and

[TABLE]

For a fixed $j\leq n$, let $\mathcal{A}$ be the set of all subsets of $j$ indices $A\subseteq[n]$. Consider a fixed $A\in\mathcal{A}$ and a word $u$ of length $j$ with letters less than $r$. If a word $W\in\Omega_{r}$ has length $n$, there are $r^{n-j}$ possible fillings of the indices in $[n]\setminus A$ and there are $f_{r}(j)$ ways to fill the indices of $A$ so that $A$ is complete in $W$.

By Lemma 5 we have

[TABLE]

By the one-to-one correspondence with complete indices $A$ in $W$ of size $j$ with vertices in $\gamma(W)$ exactly distance $j$ from the root, the expectation of $D^{(r)}_{j}$ is

[TABLE]

For $r\geq j$,

[TABLE]

and $\mathbf{E}_{\alpha,r}[D^{(r)}_{j}]=\sum_{n\geq j}\mathbf{E}[D^{(r)}_{j}\mathbf{1}_{|W|=n}]$, so

[TABLE]

of Theorem 1.

From (7), $\mathbf{E}_{\alpha,r}[D^{(r)}_{j}]=\alpha^{j}$ for $j\leq r$ and $\mathbf{E}_{\alpha,r}[D^{(r)}]=\sum_{j=0}^{r}\alpha^{j}$. Then $\lim_{r\to\infty}D^{(r)}=D$ and by Monotone Convergence Theorem

[TABLE]

∎

Expected number of leaves

For a set of indices $A$ of size $j$ that are complete in $W$, let $X$ denote the word obtained after bumping every index in $A$. The vertex labelled with $X$ is a leaf if it contains no bump-able indices, that is $X$ has no [math]s. Let $a_{0}=0$ and $a_{j+1}=|W|+1.$ For $0\leq i\leq j$, an index $b_{i}\in(a_{i},a_{i+1})$ is bump-able in $X$ if and only if $W_{b_{i}}=j-i.$ If $r\leq j$ and $i\leq j-r$, $W_{b_{i}}<r\leq j-i$ and hence $b_{i}$ cannot be bump-able. Otherwise if $i>j-r$, there are $r-1$ choices for $W_{b_{i}}$ so that $b_{i}$ is not bump-able.

Let $\ell(r,n,A)$ denote the number words, $w$ of length $n$ in $\Omega_{r}$ such that $A$ corresponds to a leaf in $\gamma(w)$. There are $f_{r}(j)$ possible ways to fill in the indices of $A$. For $r\leq j$,

[TABLE]

For $j<r$ this simplifies to

[TABLE]

Thus for $j<r$ we have

[TABLE]

For $j<r$ the expectation of $U^{(r)}_{j}\mathbf{1}_{\{|W|=n\}}$ is

[TABLE]

Summing over $n\geq j$ gives

[TABLE]

of Theorem 2.

From (12), $\mathbf{E}_{\alpha,r}[U^{(r)}_{j}]=e^{-\alpha}\alpha^{j}$ for $j<r$ and $\mathbf{E}_{\alpha,r}[U^{(r)}]=\sum_{j=0}^{r-1}e^{-\alpha}\alpha^{j}$. Then $\lim_{r\to\infty}U^{(r)}=U$ and by Monotone Convergence Theorem

[TABLE]

∎

6 Expectation of $D^{2}$

For $a,b,c,m\geq 0$ let $n=a+b+c+m$. Let $\mathcal{B}(a,b,c,m)$ be the set of all ordered pairs of subsets of $[n]$, $(A,B)$, such that $|A\setminus B|=a$, $|B\setminus A|=b$, and $|A\cap B|=c$ and let $\mathcal{B}(a,b,c)=\bigcup_{m}\mathcal{B}(a,b,c,m)$. We denote the set of distinct subwords $u$ on the indices $A\cup B$ such that and both $u_{A}$ and $u_{B}$ are complete by $\chi_{r}(A,B)$. The size of $\chi_{r}(A,B)$ is denoted by $x_{r}(A,B)$ and only depends on the relative order of $A$ and $B$. Suppose $(A,B)\in\mathcal{B}(a,b,c)$. For both subwords to be complete, each index $a_{i}\in A\setminus B$ must have letters strictly less than $\min(a+c-i,r)$, each index $b_{j}\in B\setminus A$ must have letters strictly less than $\min(b+c-j,r)$, and each index $a_{i}=b_{j}\in A\cap B$ must have letters strictly less than $\min(a+c-i,b+c-j,r)$. Thus

[TABLE]

The following lemma provides uniform bounds of $x_{r}(A,B)$ for all $(A,B)\in\mathcal{B}(a,b,c)$.

Lemma 7**.**

Fix $a,b,c$ and $r\geq 0$. For $(A,B)\in\mathcal{B}(a,b,c)$, if $a\leq b$, then

[TABLE]

Otherwise if $a>b$, then

[TABLE]

Proof.

For a fixed $a,b,c$ and $r$, $x_{r}({A,B})$ will reach its minimum value over $\mathcal{B}(a,b,c)$ when the product in the denominator is maximized in the right hand side of (14). The denominator of $x_{r}(A,B)$ is maximized when every index in $A\cap B$ is less than every index in $A\cup B\setminus A\cap B$ so $A\cap B=\{a_{1}=b_{1},\cdots,a_{c}=b_{c}\}$. In this case for $a\leq b$ the denominator of the right hand side of (14) is given by

[TABLE]

and

[TABLE]

Otherwise for $a>b$

[TABLE]

For the other direction $x_{r}(A,B)$ is maximized when the denominator in the right-hand side of (14) is minimized. This occurs when every index in $A\cap B$ is greater than every index in $A\cup B\setminus A\cap B$. In this case,

[TABLE]

These bounds on $x_{r}(A,B)$ will give us bounds on $\mathbf{E}_{\alpha}[D^{2}].$ Let $V_{r}=1+\sum_{j=1}^{\infty}D^{(r)}_{j}.$ For a fixed set of indices $A\in\mathbb{Z}_{+}$ let $\mathbf{1}_{A}(W)$ denote the indicator function that is $1$ if $W_{A}$ is complete and [math] if $W_{A}$ is not complete or $A$ is not a subset of indices of $W$. Then

[TABLE]

with $\lim_{r\to\infty}V_{r}=D$. We also have

[TABLE]

For a fixed pair $(A,B)\in\mathcal{B}(a,b,c,m)$, using Lemma 5 we have

[TABLE]

The value of $x_{r}(A,B)$ depends on $(A,B)$ but the upper and lower bounds from Lemma 7 only depend on $a,b,$ and $c$. Thus we have bounds of (16) that are uniform for all $(A,B)\in\mathcal{B}(a,b,c,m)$. For each $m$ the size of $\mathcal{B}(a,b,c,m)$ is ${a+b+c+m\choose a,b,c,m}=\frac{(a+b+c+m)!}{a!b!c!m!}$. Thus

[TABLE]

Summing over $m\geq 0$ in (17) gives the lower bound

[TABLE]

Similarly for the upper bound we have

[TABLE]

of Theorem 3.

In this section we make repeated use of the identity

[TABLE]

See Wilf (2006) for a variety of similar identities.

By Fatou’s Lemma $\lim_{r\to\infty}\mathbf{E}_{\alpha,r}[V_{r}^{2}]\leq\mathbf{E}_{\alpha}[\lim_{r\to\infty}V_{r}^{2}]=\mathbf{E}_{\alpha}[D^{2}]$ so

[TABLE]

The right hand side of (20) can be simplified further. Suppose $1/2<\alpha<1.$ Then

[TABLE]

There is an issue when $\alpha=1/2$ in (22) and (23). But in this case $\frac{\alpha}{1-\alpha}=1$ in (21), so (22) becomes $\sum_{b\geq 0}\frac{b\alpha^{b}}{1-\alpha},$ which is finite. Otherwise (23) diverges precisely when $\alpha^{2}/(1-\alpha)\geq 1$ which occurs if $({\sqrt{5}-1})/{2}\leq\alpha<1.$ For the other direction we have

[TABLE]

The last line (24) converges when ${\alpha}/{(1-\alpha)^{2}}<1,$ which occurs when $0<\alpha<(3-\sqrt{5})/2.$ ∎

As $\alpha$ increases from $(3-\sqrt{5})/2$ to $({\sqrt{5}-1})/{2}$ a phase transition occurs where $\mathbf{E}_{\alpha}[D^{2}]$ becomes infinite. With a more precise analysis of the size of $x_{r}(A,B)$ that depends more closely on the relative order of $A$ and $B$, one might be able to obtain the exact location where this phase transition occurs.

Acknowledgements

We wish to express thanks to Tobias Johnson and Anne Schilling for useful discussions.