Mutual information decay for factors of IID

TL;DR

This paper investigates how mutual information between vertices in factor of i.i.d. processes on regular trees decays with distance, revealing a faster decay rate than previously established bounds.

Contribution

It provides new upper bounds for mutual information decay at arbitrary distances and uncovers a faster decay rate for fixed processes compared to general bounds.

Findings

Mutual information decays as (d-1)^{-k/2} for vertices at distance k.

For fixed processes, the decay rate is essentially (d-1)^{-k}.

Bounds are sharp, indicating precise characterization of decay rates.

Abstract

This paper is concerned with factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We study the mutual information of the values on two given vertices. If the vertices are neighbors (i.e., their distance is $1$), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$, of order $(d-1)^{-k/2}$. Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process the rate of decay of the mutual information is much faster, essentially of order $(d-1)^{-k}$.

Equations

$$H(X_{u},X_{v})\geq\frac{2(d-1)}{d}H(X_{v})\mbox{, where $uv$ is an edge.}$$

$$\frac{I(X_u;X_v)}{H(X_v)}\le \frac{2}{d}.$$

$$\frac{I(X_{u};X_{v})}{H(X_{v})}\leq\begin{cases}\frac{2}{d(d-1)^{l}}&\mbox{ if $k=2l+1$ is odd,}\\ \frac{1}{(d-1)^{l}}&\mbox{ if $k=2l$ is even.}\end{cases}$$

$$I(X_u;X_v)\le \frac{|M|(k+1{)}^2}{(d-1{)}^k},$$

$$H(X)\text{vbox}..=\sum _{i=1}^m-p_i\log (p_i).$$

$$I(X;Y)\text{vbox}..=H(X)+H(Y)-H(X,Y).$$

$$I(X;Y)=H(X)-H(X|Y)=H(Y)-H(Y|X),$$

$$H(X|Y)=\sum _{j=1}^n\mathbb{P}(Y=y_j)\sum _{i=1}^m-\mathbb{P}(X=x_i|Y=y_j)\log \mathbb{P}(X=x_i|Y=y_j),$$

$$H(X|Y)=\mathbb{E}\sum _{i=1}^m-f_i(Y)\log f_i(Y).$$

$$(\gamma \cdot f)(s)\text{vbox}..=f({\gamma }^{-1}\cdot s)\forall s\in S.$$

$$\frac{1}{r}\sum _{i=1}^{r}H(X_g,X_{g{a}_i})\ge \frac{2r-1}{r}H(X_g),$$

$$\frac{1}{r}\sum _{i=1}^r\frac{I(X_g;X_{g{a}_i})}{H(X_g)}\le \frac{1}{r}.$$

$$G=\underset{r}{\underbrace{\mathbb{Z}\ast \cdots \ast \mathbb{Z}}}\ast \underset{t}{\underbrace{{\mathbb{Z}}_2\ast \cdots \ast {\mathbb{Z}}_2}}=⟨a_1,\dots ,a_r,a_{r+1},\dots ,a_{r+t}|a_{r+1}^2=\cdots =a_{r+t}^2=e⟩.$$

$$G=F_r=\underset{r}{\underbrace{\mathbb{Z}\ast \cdots \ast \mathbb{Z}}}=⟨a_1,\dots ,a_r⟩.$$

$$G=F_r\ast {\mathbb{Z}}_2=\underset{r}{\underbrace{\mathbb{Z}\ast \cdots \ast \mathbb{Z}}}\ast {\mathbb{Z}}_2=⟨a_1,\dots ,a_r,a_{r+1}|a_{r+1}^2=e⟩.$$

$$G=\underset{d}{\underbrace{{\mathbb{Z}}_2\ast \cdots \ast {\mathbb{Z}}_2}}=⟨a_1,\dots ,a_d|a_1^2=\cdots =a_d^2=e⟩.$$

$$[0,1{]}^H\stackrel{\phi \times \phi \times \cdots }{\to }\{0,1{\}}^{H\times \mathbb{N}}\stackrel{\beta \times \beta \times \cdots }{\to }\{0,1{\}}^{H\times T\times \mathbb{N}}\stackrel{\alpha \times \alpha \times \cdots }{\to }\{0,1{\}}^{G\times \mathbb{N}}\stackrel{\psi \times \psi \times \cdots }{\to }[0,1{]}^G.$$

$$k=2l+1\ge 3$$

$$k=2l$$

$$[0,1{]}^H\stackrel{\varrho }{\to }[0,1{]}^G\stackrel{F}{\to }{M}^G\stackrel{{\pi }_H}{\to }{M}^H,$$

$$S\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left\{s\in G:\mbox{the reduced form of $s$ is a palindrome and has length $2l+1$}\right\}.$$

$$B_0\text{vbox}..=\left\{a_i{a}_1:2\le i\le d\right\}$$

$$B_0^{-1}=\left\{a_1{a}_i:2\le i\le d\right\}.$$

$${\phi }_j(b_l)\cdots {\phi }_j(b_2)\underset{a_{j+1}}{\underbrace{{\phi }_j(b_1)}}\underset{a_1}{\underbrace{b_1}}{b}_2\cdots b_l.$$

$$\left|\mathrm{corr}(X_u,X_v)\right|\le \left(k+1-\frac{2k}{d}\right){\left(\frac{1}{\sqrt{d-1}}\right)}^k\text{mbox},wherek=\mathrm{dist}(u,v),$$

$$\left|\operatorname{corr}\big{(}f(X_{1}),f(X_{2})\big{)}\right|\leq\alpha\mbox{ provided that $f(X_{1})$ has finite variance.}$$

$$\left|\operatorname{corr}\big{(}f_{1}(X_{1}),f_{2}(X_{2})\big{)}\right|\leq\alpha\mbox{ provided that $f_{1}(X_{1})$ and $f_{2}(X_{2})$ have finite variances.}$$

$$I(X;Y)=H(X)-H(X|Y)\le (m-1){\alpha }^2,$$

$$g_A(y)\text{vbox}..=\mathbb{P}(A|Y=y)-\mathbb{P}(A)=\mathbb{P}(A|Y=y)-p.$$

$$\operatorname{corr}\big{(}f(X),g_{A}(Y)\big{)}=\frac{\sqrt{\mathbb{E}g_{A}(Y)^{2}}}{\sqrt{p(1-p)}}.$$

Full text

The first author was supported by NKFIH (National Research, Development and Innovation Office) grant PD 121107. The second author was supported by Marie Skłodowska-Curie Individual Fellowship grant no. 661025 and the MTA Rényi Institute “Lendület” Groups and Graphs Research Group.

1. Introduction

For an integer $d\geq 3$ let $T_{d}$ denote the $d$-regular tree: the (infinite) connected graph with no cycles and with each vertex having exactly $d$ neighbors.

This paper deals with factor of i.i.d. processes on $T_{d}$. First we give an informal definition: independent and identically distributed (say $[0,1]$ uniform) random labels are assigned to the vertices of $T_{d}$, then each vertex gets a new label that depends on the labeled rooted graph as seen from that vertex, all vertices “using the same rule”.

For a formal definition, let $V(T_{d})$ denote the vertex set and $\operatorname{Aut}(T_{d})$ the automorphism group of $T_{d}$. Suppose that $M$ is a measurable space. (In most cases $M$ will be either a discrete set or $\mathbb{R}$.) A measurable function $F\colon[0,1]^{V(T_{d})}\to M^{V(T_{d})}$ is said to be an $\operatorname{Aut}(T_{d})$-factor (or factor in short) if it is $\operatorname{Aut}(T_{d})$-equivariant, that is, it commutes with the natural $\operatorname{Aut}(T_{d})$-actions. Given an i.i.d. process $Z=\left(Z_{v}\right)_{v\in V(T_{d})}$ on $[0,1]^{V(T_{d})}$, applying $F$ yields a factor of i.i.d. process $X=F(Z)$, which can be viewed as a collection $X=\left(X_{v}\right)_{v\in V(T_{d})}$ of $M$-valued random variables. It follows immediately from the definition that the distribution of $X$ is invariant under the action of $\operatorname{Aut}(T_{d})$; in particular, each $X_{v}$ has the same distribution. Factors of i.i.d. are also studied by ergodic theory (under the name of factors of Bernoulli shifts), see Section 2 for details.

One of the reasons why factor of i.i.d. processes have attracted a growing attention in recent years is that they give rise to certain randomized local algorithms. Suppose that we have a finite $d$-regular graph that locally looks like $T_{d}$, that is, around most vertices the neighborhoods are trees (up to some large radius). Then i.i.d. labels can be put on the vertices and a given factor mapping can be applied (approximately) at each vertex, yielding a randomized algorithm on the finite graph. The distribution of the random output of this algorithm is described locally by the factor of i.i.d. process. See [10, 11, 12, 13] for how such local algorithms can be used to obtain large independent sets. (Whether a graph is locally tree-like is related to the number of cycles. The girth of a graph is the length of its shortest cycle. When we say that a finite graph has large essential girth, we mean that the number of short cycles is small compared to the number of vertices. Around most vertices of such a graph the neighborhoods are trees up to a large radius. Note that random regular graphs have large essential girth with high probability.)

The starting point of our investigations is the following entropy inequality which holds for any factor of i.i.d. process $X$ with a finite state space $M$:

[TABLE]

Here $H(X_{v})$ is the (Shannon) entropy of the discrete random variable $X_{v}$, and $H(X_{u},X_{v})$ stands for the joint entropy of $X_{u}$ and $X_{v}$, see Section 2.1 for the definitions. (Because of the $\operatorname{Aut}(T_{d})$-invariance the distribution of $X_{v}$ is the same for each vertex $v$. Similarly, the joint distribution of $(X_{u},X_{v})$ is the same for any edge $uv$.) Rahman and Virág proved (1) in a special setting [16]. A full and concise proof was given by Backhausz and Szegedy in [2]; see also [15]. The counting argument behind this inequality actually goes back to a result of Bollobás on the independence ratio of random regular graphs [6]. As we will see in Section 2.4, a more general version of (1) can also be found implicitly (for even $d$) in Lewis Bowen’s work on free group actions [8].

Entropy inequalities played a central role in a couple of remarkable results recently: the Rahman-Virág result [16] about the maximal size of a factor of i.i.d. independent set on $T_{d}$ and the Backhausz-Szegedy result [3] on eigenvectors of random regular graphs.

The inequality (1) can also be expressed as an upper bound for the mutual information of two neighboring vertices $u$ and $v$:

[TABLE]

Recall that the mutual information $I(X_{u};X_{v})$ is defined as $H(X_{u})+H(X_{v})-H(X_{u},X_{v})$ and can be viewed as (the expected value of) the information gained about one of the random variables knowing the other one. In our case the random variables are identically distributed, therefore they have the same entropy $H(X_{u})=H(X_{v})$. Dividing the mutual information by this entropy results in a normalized mutual information which measures the amount of shared information proportional to the total amount of information. This ratio is always between [math] and $1$, and being close to [math] intuitively means that the random variables are “almost independent”. (It is reasonable to normalize the mutual information this way, see Example 2.2.)

A natural question arises: what can be said about the mutual information of two vertices $u$ and $v$ at distance $k$? One expects that the mutual information tends to [math] as the distance grows. But what is the rate of decay? We get very different answers depending on how the question is posed exactly.

First let us consider the problem for a fixed $k\geq 1$, that is, we look for a universal upper bound for the normalized mutual information $I(X_{u};X_{v})/H(X_{v})$ that holds for any factor of i.i.d. process with a finite state space $M$. The following bounds are obtained.

Theorem 1**.**

Let $M$ be a finite state space and $d\geq 3$. For any $u,v\in V(T_{d})$ at distance $k$ and for any factor of i.i.d. process $X$ on $M^{V(T_{d})}$ we have

[TABLE]

These bounds are the best possible in the sense that for any fixed $k$ there exist factor of i.i.d. processes for which the normalized mutual information tends to the bound above.

According to (3), the normalized mutual information for distance $k$ is (at most) of order $(1/\sqrt{d-1})^{k}$, and this is sharp. However, it turns out that there does not exist a single factor of i.i.d. process that would show the sharpness of the bound for all $k$ at once. In fact, for any fixed process the mutual information decays at a much faster rate, basically of order $1/(d-1)^{k}$.

Theorem 2**.**

Let $M$ be a finite state space and $d\geq 3$. If $X=\left(X_{v}\right)_{v\in V(T_{d})}$ is a factor of i.i.d. process on $M^{V(T_{d})}$, then

[TABLE]

where $|M|$ denotes the cardinality of $M$ (number of states).

This bound is essentially sharp, see Example 5.4.

Motivation

Our motivation to study this problem is multi-fold. On the one hand, many aspects of independence in factors of i.i.d. have been studied earlier (e.g. correlation for real-valued processes or triviality of various tail $\sigma$-algebras). Our goal was to get a quantitative result about how much independence these processes exhibit when $M$ is finite. Mutual information has the advantage over correlation that the latter only detects linear dependence. On the other hand, we aimed to obtain new entropy inequalities. The edge-vertex inequality (1) and its blow-ups (where both the vertex $v$ and the edge $uv$ are replaced with all the vertices in their respective $R$-radius neighborhoods) have a number of applications already. Theorem 1 is a generalization of (1), and as such one expects it will provide further applications.

Proof methods

To prove Theorem 1 we will consider the $d$-regular tree $T_{d}$ as the Cayley graph of different groups $G$ depending on the parity of $d$ and $k$. When $k$ is even, we will use the free product $G=\mathbb{Z}_{2}\ast\cdots\ast\mathbb{Z}_{2}$. When $k$ is odd, either $G=\mathbb{Z}\ast\cdots\ast\mathbb{Z}$ (for even $d$) or $G=\mathbb{Z}\ast\cdots\ast\mathbb{Z}\ast\mathbb{Z}_{2}$ (for odd $d$) will be used. In each case we will try to find as many elements in $G$ as possible such that they freely generate a subgroup and each element has length $k$ (w.r.t. the corresponding word metric in $G$). In other words, we will look for a maximum-rank free subgroup $H\leq G$ that has a generating set consisting of elements with length $k$. Once we have such a free subgroup $H$, Theorem 1 will follow from a more general version of the edge-vertex entropy inequality (Theorem 2.3). This inequality is known from Lewis Bowen’s work on free group actions, namely it is equivalent to the fact that the $f$-invariant is non-negative for factors of the Bernoulli shift [8].

Theorem 2 will be deduced from the correlation decay result of Backhausz, Szegedy, and Virág [4], which says that for a real-valued factor of i.i.d. process ($M=\mathbb{R}$) the correlation of two vertices $u$ and $v$ at distance $k$ is (at most) of order $1/(\sqrt{d-1})^{k}$. In the case of a finite state space $M$, by assigning a real number to each state we can replace our original process with a real-valued one. Consequently, for any assignment $M\to\mathbb{R}$ the correlation bound tells us something about the joint distribution of $X_{u}$ and $X_{v}$ (for the original process). The idea is to try to find suitable assignments that yield a good bound on the mutual information of $X_{u}$ and $X_{v}$.

Outline of the paper

The rest of the paper is structured as follows. In Section 2 we go through basic definitions and explain the more general entropy inequality we will need to prove the universal bound. The proofs of Theorem 1 and 2 are given in Section 3 and 4, respectively. Finally, in Section 5 we present examples showing that the above theorems are (essentially) sharp.

Acknowledgments

We are grateful to Ágnes Backhausz, Balázs Szegedy, Bálint Virág, and Máté Vizer for fruitful discussions on the topic. We would also like to thank the anonymous referee for many valuable comments and suggestions.

2. Preliminaries

2.1. Entropy and mutual information

Let $X$ be a discrete random variable taking $m$ distinct values with probabilities $p_{1},\ldots,p_{m}$. Then the Shannon entropy of $X$ is defined as

[TABLE]

Given two discrete random variables $X$ and $Y$, $(X,Y)$ can be considered as a discrete random variable itself, and its entropy is denoted by $H(X,Y)$. (This is often called the joint entropy of $X$ and $Y$.) One can define the mutual information of $X$ and $Y$ by

[TABLE]

Another way to define mutual information is via conditional entropies:

[TABLE]

where the conditional entropy $H(X|Y)=H(X,Y)-H(Y)$ can be expressed as the expectation (in $Y$) of the entropy of the (conditional) distribution of $X$ conditioned on $Y$, that is,

[TABLE]

where $x_{1},\ldots,x_{m}$ and $y_{1},\ldots,y_{n}$ denote the values taken by $X$ and $Y$, respectively. In other words, if $f_{i}$ denotes the mapping $y\mapsto\mathbb{P}(X=x_{i}|Y=y)$, then

[TABLE]

2.2. Factors of i.i.d.

Although the results of this paper concern $\operatorname{Aut}(T_{d})$-factors, we will need to use the notion of factors in a more general setting. Suppose that a group $\Gamma$ acts on a countable set $S$. Then $\Gamma$ also acts on the space $M^{S}$ for a set $M$: for any function $f\colon S\to M$ and for any $\gamma\in\Gamma$ let

[TABLE]

First we define the notion of factor maps.

Definition 2.1**.**

Let $M_{1},M_{2}$ be measurable spaces and $S_{1},S_{2}$ countable sets with a group $\Gamma$ acting on both. A measurable mapping $F\colon M_{1}^{S_{1}}\to M_{2}^{S_{2}}$ is said to be a $\Gamma$-factor if it is $\Gamma$-equivariant, that is, it commutes with the $\Gamma$-actions.

By an invariant (random) process on $M^{S}$ we mean an $M^{S}$-valued random variable (or a collection of $M$-valued random variables) whose (joint) distribution is invariant under the $\Gamma$-action. An important class of invariant processes is factor of i.i.d. processes defined as follows. Suppose that $Z_{s}$, $s\in S_{1}$, are independent and identically distributed $M_{1}$-valued random variables. We say that $Z=\left(Z_{s}\right)_{s\in S_{1}}$ is an i.i.d. process on $M_{1}^{S_{1}}$. Given a $\Gamma$-factor $F\colon M_{1}^{S_{1}}\to M_{2}^{S_{2}}$, $X\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=F(Z)$ is a factor of the i.i.d. process $Z$. It can be regarded as a collection of $M_{2}$-valued random variables: $X=\left(X_{s}\right)_{s\in S_{2}}$.

In fact, all this can be viewed in the context of ergodic theory. An invariant process in the above sense gives rise to a dynamical system over $\Gamma$: the group $\Gamma$ acts by measure-preserving transformations on the measurable space $M^{S}$ equipped with a probability measure (the distribution of the invariant process). An i.i.d. process simply corresponds to a (generalized) Bernoulli shift. Therefore factor of i.i.d. processes are essentially factors of Bernoulli shifts. Classical ergodic theory ($\mathbb{Z}$-factors) have the largest literature and the most complete theory but $\Gamma$-factors have also been thoroughly investigated for general $\Gamma$.

For amenable group actions (the Kolmogorov-Sinai) entropy serves as a complete invariant (for isomorphism of Bernoulli shifts). As for the nonamenable case, Ornstein and Weiss asked whether all Bernoulli shifts are isomorphic over a nonamenable group [14]. This remained open until the breakthrough results of Lewis Bowen: he answered the question negatively by introducing the $f$-invariant for free group actions [7] and the $\Sigma$-entropy for actions of sofic groups [9]. In another paper he showed that the $f$-invariant is essentially a special case of the $\Sigma$-entropy which has the consequence that the $f$-invariant is non-negative for factors of the Bernoulli shift [8, Corollary 1.8]. We will need this fact in the form of an entropy inequality, see (7) below.

2.3. Factors on $T_{d}$

The main results of this paper (Theorem 1 and 2) are concerned with factor of i.i.d. processes on $T_{d}$. This corresponds to the case when $\Gamma$ is the automorphism group $\operatorname{Aut}(T_{d})$ of the $d$-regular infinite tree $T_{d}$ and $S$ is the vertex set $V(T_{d})$.

When we say factor of i.i.d. process, we should also specify which i.i.d. process we have in mind (that is, specify $M_{1}$ and a probability distribution on it). By default we will work with the uniform $[0,1]$ measure (i.e., the Lebesgue measure on $[0,1]$). In fact, as far as the class of factor processes is concerned, it does not really matter which i.i.d. process we consider. For example, for $\{0,1\}$ with the uniform distribution we get the same class of factors as for the uniform $[0,1]$ measure. This follows from the fact that these two i.i.d. processes are $\operatorname{Aut}(T_{d})$-factors of each other [5].

Note that a factor of i.i.d. process $X$ on $T_{d}$ is $\operatorname{Aut}(T_{d})$-invariant. Therefore each $X_{v}$ has the same distribution. Moreover, the joint distribution of $X_{u}$ and $X_{v}$ (and hence their correlation or mutual information) depends only on the distance between $u$ and $v$.

One of our goals in this paper is to find a universal upper bound for the mutual information $I(X_{u};X_{v})$ that holds for any factor of i.i.d. process $X$. The next example, where a tuple of independent copies of the same factor of i.i.d. process is considered, shows that this goal is plausible only if we normalize $I(X_{u};X_{v})$ in some way. That is why we introduced the normalized mutual information $I(X_{u};X_{v})/H(X_{v})$.

Example 2.2**.**

Given a factor of i.i.d. process $X=F(Z)$ with a finite state space $M$ there exists a factor of i.i.d. process $Y=(Y^{1},\ldots,Y^{n})$ with state space $M^{n}=M\times\cdots\times M$ such that each $Y^{i}=\left(Y^{i}_{v}\right)_{v\in V(T_{d})}$ is an independent copy of $X$. (The point is that one can take $n$ independent copies $Z^{1},\ldots,Z^{n}$ of the i.i.d. process $Z$ and apply $F$ to each $Z^{i}$ to get $Y^{i}$. It is easy to see that $(Z^{1},\ldots,Z^{n})$ can be obtained as a factor of $Z$. Therefore the process $Y$ is also a factor of $Z$.) If we take $n$ copies of $X$ as described above, then each entropy and mutual information gets multiplied by $n$. On the other hand, the normalized mutual information (corresponding to two given vertices $u$ and $v$) is the same for $X$ and $Y$.

2.4. $F_{r}$-factors

The other case that will be of particular interest for us is when $\Gamma$ is the free group $F_{r}$ of some rank $r$. We can set $S=\Gamma=F_{r}$ and consider the natural action of $F_{r}$ on itself. Similarly as for $\operatorname{Aut}(T_{d})$-factors, we use the uniform $[0,1]$ measure for the i.i.d. process. Using other measures would result in the same class of factor processes.

This is actually a broader class than the class of $\operatorname{Aut}(T_{d})$-factors (for $d=2r$). If $d=2r$, we can think of $T_{d}$ as the Cayley graph of $F_{r}$ with respect to a symmetric generating set $\{a_{1}^{\pm 1},\ldots,a_{r}^{\pm 1}\}$. That is, $V(T_{d})=F_{r}$ and a vertex $g$ is incident to vertices of the form $ga_{i}^{\pm 1}$. Then $F_{r}$ acts on $V(T_{d})=F_{r}$ (from the left) via automorphisms of this Cayley graph. So if we identify the elements of $F_{r}$ with these automorphisms, then $F_{r}$ becomes a subgroup of $\operatorname{Aut}(T_{d})$, and consequently being $\operatorname{Aut}(T_{d})$-equivariant is a stronger condition than being $F_{r}$-equivariant. In other words, every $\operatorname{Aut}(T_{d})$-factor is an $F_{r}$-factor as well.

For a general $F_{r}$-factor of i.i.d. we only have $F_{r}$-invariance (but not necessarily $\operatorname{Aut}(T_{d})$-invariance). It is still true that each $X_{g}$ has the same distribution. As for the distribution of edges, however, $(X_{g},X_{ga_{i}^{\pm 1}})$ might have different distributions for different $a_{i}^{\pm 1}$.

The following entropy inequality, which plays a central role in our proof of Theorem 1, easily follows from the fact that the $f$-invariant of a factor of a Bernoulli shift is non-negative [8].

Theorem 2.3**.**

Let $\Gamma=\left\langle a_{1},\ldots,a_{r}\right\rangle$ be a free group of rank $r\geq 2$. If $X=\left(X_{g}\right)_{g\in\Gamma}$ is a $\Gamma$-factor of the i.i.d. process on $[0,1]^{\Gamma}$, then for a fixed $g\in\Gamma$ we have

[TABLE]

or equivalently:

[TABLE]

Remark 2.4**.**

This is more general than the edge-vertex entropy inequality (1) for $\operatorname{Aut}(T_{d})$-factors for $d=2r$. Indeed, given an $\operatorname{Aut}(T_{d})$-factor, it is also an $F_{r}$-factor, but with the extra property that the distributions of edges are the same.

3. The universal bound

In this section $G$ will denote the free product of $r$ copies of $\mathbb{Z}$ and $t$ copies of $\mathbb{Z}_{2}$ for different values of $r$ and $t$:

[TABLE]

Let $A$ denote the set $\{a_{1}^{\pm 1},\ldots,a_{r}^{\pm 1},a_{r+1},\ldots,a_{r+t}\}$. First we define the word metric on $G$ with respect to $A$. We will refer to the elements of $A$ as letters and to products of these elements as words. An element $g\in G$ can be represented by many words but for each $g$ there exists a unique shortest representing word. Actually, starting with any word representing $g$, by performing all possible cancellations in that product one always gets the shortest representing word that we will call the reduced form. We define the length of $g$ as the length of this reduced form. (As for the unit element $e$ of $G$, it is represented by the empty product, and hence the length of $e$ is [math].)

Note that the Cayley graph of $G$ with respect to $A$ is $T_{d}$ for $d=2r+t$. (That is, $V(T_{d})=G$ and a vertex $g$ is incident to vertices of the form $gh$, $h\in A$.) The word metric on $G$ (w.r.t. $A$) actually coincides with the graph distance on this Cayley graph.

Our goal is to apply the inequality (7–8) for free subgroups of $G$. To obtain a result about vertices at distance $k$ in $T_{d}$ we will need a free subgroup $H$ that is generated by elements of length $k$. The higher the rank of our subgroup, the better inequality we get. Therefore we need to find as many elements of length $k$ as possible such that they freely generate a subgroup. (Although we will not need this fact, we mention that when we have the maximal possible number of elements, the generated subgroup has finite index.)

Lemma 3.1**.**

Let $d=2r$ and let

[TABLE]

Then for any odd integer $k=2l+1$ there exists a free subgroup $H\leq G$ of rank $d(d-1)^{l}/2$ that is generated freely by elements of length $k$ (in the corresponding word metric).

Lemma 3.2**.**

Let $d=2r+1$ and let

[TABLE]

Then for any odd integer $k=2l+1$ with $l\geq 1$ there exists a free subgroup $H\leq G$ of rank $d(d-1)^{l}/2$ that is generated freely by elements of length $k$ (in the corresponding word metric).

Lemma 3.3**.**

Let $d\geq 3$ be arbitrary and let

[TABLE]

Then for any even integer $k=2l$ there exists a free subgroup $H\leq G$ of rank $(d-1)^{l}$ that is generated freely by elements of length $k$ (in the corresponding word metric).

Before we prove the above lemmas, let us show how Theorem 1 follows. We start with a technical lemma.

Lemma 3.4**.**

Suppose that $H$ is a subgroup of a countable group $G$. Let us equip the spaces $[0,1]^{H}$ and $[0,1]^{G}$ with the product of uniform $[0,1]$ measures. Then there exists a $[0,1]^{H}\to[0,1]^{G}$ mapping that is measure-preserving and $H$-equivariant.

Proof.

Let us fix measure-preserving mappings $\varphi\colon[0,1]\to\{0,1\}^{\mathbb{N}}$ and $\psi\colon\{0,1\}^{\mathbb{N}}\to[0,1]$, where $\{0,1\}$ is equipped with the (discrete) uniform distribution.

Let us also fix a set $T$ that contains exactly one element of each right $H$-coset, meaning that $(h,t)\mapsto ht$ defines a bijection $H\times T\to G$. Using the trivial $H$-action on $T$ and the natural (left) $H$-actions on $H$ and $G$ the above bijection will clearly be $H$-equivariant. This induces an $H$-equivariant mapping $\alpha\colon\{0,1\}^{H\times T}\to\{0,1\}^{G}$.

Since $T$ is either finite or countably infinite, $T\times\mathbb{N}$ has the same cardinality as $\mathbb{N}$, so we can fix a bijection between these sets as well. This bijection yields a measure-preserving mapping $\beta\colon\{0,1\}^{\mathbb{N}}\to\{0,1\}^{T\times\mathbb{N}}$.

Combining the above mappings we get the following:

[TABLE]

Each of the above mappings clearly preserves measure and commutes with the $H$-actions. ∎

Proof of Theorem 1.

For $k=1$ the statement of the theorem is equivalent to (2) so we may assume that $k\geq 2$. Depending on $d$ and $k$ we choose the group $G$ and the positive integer $r^{\prime}$ as follows:

[TABLE]

Let $A\subset G$ still denote the generating set described at the beginning of this section. Recall that the Cayley graph of $G$ with respect to $A$ is $T_{d}$ so from this point on $V(T_{d})$ is identified with $G$. According to Lemma 3.1–3.3 in each of the above cases $G$ has a free subgroup $H$ of rank $r^{\prime}$ such that $H$ has a free generating set $S_{0}$ consisting of elements of length $k$ (in the word metric of $G$ with respect to $A$).

Now let $X=\left(X_{v}\right)_{v\in G}$ be a factor of i.i.d. process over $V(T_{d})=G$ with a finite state space $M$. This means that there exists an $\operatorname{Aut}(T_{d})$-factor mapping $F\colon[0,1]^{G}\to M^{G}$ such that $X=F(Z)$ where $Z$ is the i.i.d. process on $[0,1]^{G}$. According to Lemma 3.4 there exist an $H$-equivariant mapping $\varrho\colon[0,1]^{H}\to[0,1]^{G}$ such that $Z=\varrho(\tilde{Z})$ where $\tilde{Z}$ is an i.i.d. process on $[0,1]^{H}$.

By $\pi_{H}$ we denote the projection $M^{G}\to M^{H}$. We have the following situation:

[TABLE]

where all three mappings are $H$-equivariant, and hence their composition is an $H$-factor mapping. This means that if we consider $X$ over $H$, then we get an $H$-factor of i.i.d. process: $\left(X_{h}\right)_{h\in H}=\pi_{H}\circ F\circ\varrho(\tilde{Z})$. Therefore we can apply (8) to $H$ and its free generating set $S_{0}$ of size $r^{\prime}$. For any $h\in H$ and any $s\in S_{0}$, the vertices $h$ and $hs$ have distance $k$ (in the graph metric of $T_{d}$). Then, because of the $\operatorname{Aut}(T_{d})$-invariance of $X$, the normalized mutual information $I(X_{h};X_{hs})/H(X_{h})$ is the same for all $h$ and $s$. Therefore in our case the average on the left-hand side of (8) is simply equal to $I(X_{u};X_{v})/H(X_{v})$ for any $u,v\in V(T_{d})$ with $\operatorname{dist}(u,v)=k$, while the right hand side is $1/r^{\prime}$, and hence Theorem 1 follows. (The sharpness will be shown in Section 5.) ∎

It remains to prove Lemma 3.1–3.3.

Proof of Lemma 3.1.

The set of letters in this case is $A=\{a_{1}^{\pm 1},\ldots,a_{r}^{\pm 1}\}$. A word is called a palindrome if it reads the same backward as forward. Let us consider the following set of words:

[TABLE]

That is, elements of $S$ are in the form $b_{1}\cdots b_{l}b_{l+1}b_{l}\cdots b_{1}$, where $b_{i}\in A$ and $b_{i+1}\neq b_{i}^{-1}$. The number of such elements is clearly $2r(2r-1)^{l}=d(d-1)^{l}$.

The inverse of a palindrome is also a palindrome (and not the same palindrome because $G$ has no elements of order $2$). Therefore there exists $S_{0}\subset S$ with $|S_{0}|=|S|/2=d(d-1)^{l}/2$ such that $S=S_{0}\cup S_{0}^{-1}$ where $S_{0}^{-1}=\{s^{-1}\,:\,s\in S_{0}\}$. We will see that $S_{0}$ is a free generating set of a subgroup $H\leq G$ that has all the required properties.

The key observation is the following.

Claim**.**

Let $s_{1},\ldots,s_{n}$ be palindromes in $S$ such that $s_{i+1}\neq s_{i}^{-1}$ for each $i$. Then the reduced form of the product $s_{1}\cdots s_{n}$ has length at least $2l+n$ and its last $l+1$ letters are the same as those of $s_{n}$.

We prove the claim by induction. It is obvious for $n=1$. For $n\geq 1$ let us assume that the reduced form of the product $s_{1}\cdots s_{n}$ ends with the same $l+1$ letters as $s_{n}$ and let $s_{n+1}\neq s_{n}^{-1}$. This means that when we multiply the reduced form of $s_{1}\cdots s_{n}$ by $s_{n+1}$ at most $l$ letters will be cancelled out and the remaining (at least $l+1$) letters of $s_{n+1}$ will appear unchanged at the end of the product. It follows that the last $l+1$ letters of the reduced form of $s_{1}\cdots s_{n}s_{n+1}$ will be the same as those of $s_{n+1}$. We also get that $s_{1}\cdots s_{n}s_{n+1}$ is at least $(l+1)-l=1$ longer than $s_{1}\cdots s_{n}$, which completes the induction.

In particular, the product $s_{1}\cdots s_{n}$ cannot be the unit element of $G$. Therefore $S_{0}$ freely generates some subgroup $H\leq G$, the rank of which is, obviously, $|S_{0}|=d(d-1)^{l}/2$, and this is what we wanted to prove.

In fact, $H$ has finite index. (We do not need this property in this paper.) This follows from the following observation. Let $T\subset G$ denote the set of elements of length at most $l$. Then it is easy to see that every element of $G$ can be (uniquely) written in the form $s_{1}\cdots s_{n}t$, where $t\in T$, $s_{i}\in S$ and $s_{i+1}\neq s_{i}^{-1}$. ∎

Proof of Lemma 3.2.

Essentially the same proof works. Here the set of letters is $A=\{a_{1}^{\pm 1},\ldots,a_{r}^{\pm 1},a_{r+1}\}$, and one of the letters ($a_{r+1}$) has order $2$ meaning $a^{-1}_{r+1}=a_{r+1}$. However, we can still define the set $S$ of palindromes of length $k=2l+1$ for which we have $|S|=(2r+1)(2r)^{l}=d(d-1)^{l}$. The same claim as in the previous proof remains true. The only difference is that in this case $G$ has elements of order $2$. So we need to check that $S$ contains no element of order $2$, which is clearly true unless $l=0$. The rest of the proof is the same. ∎

Proof of Lemma 3.3.

In this lemma the set of letters $A=\{a_{1},\ldots,a_{d}\}$ consists of elements of order $2$. It is an easy exercise that for $l=1$ the set

[TABLE]

is a free generating set (of size $d-1$) of the subgroup of $G$ consisting of all elements of even length. Note that

[TABLE]

For $l\geq 2$ we will need to nest the $d-1$ elements of $B_{0}$ in palindrome-like words of length $2l$. First we define the mappings $\varphi_{j}\colon A\to A$: for $j\in\{1,\ldots,d-1\}$ let $\varphi_{j}$ shift the indices by $j$, that is, $\varphi_{j}(a_{i})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=a_{i+j}$. (The addition in the index is meant modulo $d$.) We will consider words of the following form: for any given $j\in\{1,\ldots,d-1\}$ and any given sequence of letters $b_{1},\ldots,b_{l}$ from $A$ such that $b_{1}=a_{1}$ and $b_{i+1}\neq b_{i}$ take the word

[TABLE]

Note that these words have length $2l$ and for the two letters in the middle we have $\varphi_{j}(b_{1})b_{1}=a_{j+1}a_{1}\in B_{0}$. We claim that the set $S_{0}$ of these $(d-1)^{l}$ words freely generates a subgroup.

The following is straightforward by induction.

Claim**.**

Let $s_{1},\ldots,s_{n}$ be words in $S\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=S_{0}\cup S_{0}^{-1}$ such that $s_{i+1}\neq s_{i}^{-1}$ for each $i$. Then the product $s_{1}\cdots s_{n}$ has the following property:

• •

if $s_{n}\in S_{0}^{-1}$, then the last $l$ letters in the reduced form of $s_{1}\cdots s_{n}$ are the same as in $s_{n}$;

• •

if $s_{n}\in S_{0}$, then the last $l+1$ letters in the reduced form of $s_{1}\cdots s_{n}$ are the same as in $s_{n}$.

It immediately follows that the length of the reduced form of the product $s_{1}\cdots s_{n}$ cannot decrease when multiplied by a new element $s_{n+1}\neq s_{n}^{-1}$. In particular, for $n\geq 1$ the product $s_{1}\cdots s_{n}$ cannot be equal to the unit element $e$. Therefore $S_{0}$ freely generates a subgroup of rank $|S_{0}|=(d-1)^{l}$. ∎

4. The rate of decay for a fixed process

We will need three ingredients to prove Theorem 2. The first one is a bound for the correlation of a pair of vertices for factor of i.i.d. processes on $\mathbb{R}^{V(T_{d})}$, which was proved by Backhausz, Szegedy, and Virág in [4]:

[TABLE]

that is, the rate of the correlation decay is essentially $1/(\sqrt{d-1})^{k}$. (Here it is assumed that $\operatorname{var}X_{v}<\infty$.)

Now suppose we have a finite state space $M$ and a factor of i.i.d. process on $M^{V(T_{d})}$. How can we make use of the above result in this case? Taking any function $f\colon M\to\mathbb{R}$ we can replace each $X_{v}$ with $f(X_{v})$ to get a factor of i.i.d. on $\mathbb{R}^{V(T_{d})}$ so that (9) can be applied. The second ingredient is the next lemma from [1] which tells us that the same bound holds if we take different real-valued functions of $X_{u}$ and $X_{v}$.

Lemma 4.1**.**

Let $(A,\mathcal{F})$ be an arbitrary measurable space. Suppose that the $(A,\mathcal{F})$-valued random variables $X_{1},X_{2}$ are exchangeable (that is, $(X_{1},X_{2})$ and $(X_{2},X_{1})$ have the same joint distribution), and that there exists a constant $\alpha\geq 0$ with the property that for any measurable $f\colon A\to\mathbb{R}$ we have

[TABLE]

Then for any measurable functions $f_{1},f_{2}\colon A\to\mathbb{R}$

[TABLE]

Proof.

The detailed proof can be found in [1, Lemma 3.2]. We include a sketch here for the sake of completeness. After rescaling we might assume that $\operatorname{var}(f_{1}(X_{1}))=\operatorname{var}(f_{2}(X_{2}))=1$. If we apply (10) to the function $f=f_{1}+f_{2}$ and also to $f=f_{1}-f_{2}$, we reach (11) after a short and simple calculation. Note that the exchangeability of $X_{1}$ and $X_{2}$ implies $\operatorname{cov}(f_{1}(X_{1}),f_{2}(X_{2}))=\operatorname{cov}(f_{1}(X_{2}),f_{2}(X_{1}))$. ∎

The final ingredient is the following lemma linking correlation to mutual information.

Lemma 4.2**.**

Let $X,Y$ be discrete random variables. Suppose that there exists a real number $\alpha\geq 0$ such that for any (real-valued) functions $f(X)$ and $g(Y)$ of $X$ and $Y$ it holds that \left|\operatorname{corr}\big{(}f(X),g(Y)\big{)}\right|\leq\alpha. Then we have

[TABLE]

where $m$ denotes the number of values $X$ can take.

Proof.

Let $A$ be an event that depends on $X$, that is, $\mathbbm{1}_{A}=f(X)$ for some function $f$. We denote the probability $\mathbb{P}(A)$ by $p$ and we set

[TABLE]

Clearly, $\mathbb{E}g_{A}(Y)=0$, and it is also easy to see that

[TABLE]

It follows that

[TABLE]

Now let us assume that $X$ takes the value $x_{i}$ with probability $p_{i}$ for $1\leq i\leq m$. We will need to use the above inequality for each event $A_{i}=\mathbbm{1}_{\{X=x_{i}\}}$, $1\leq i\leq m$. We write $g_{i}$ for the corresponding function $g_{A_{i}}$.

According to (5) the conditional entropy $H(X|Y)$ can be expressed as

[TABLE]

Now by using the inequality $\log(1+x)\leq x$ we get that

[TABLE]

Using that $\mathbb{E}g_{i}(Y)=0$ we conclude that

[TABLE]

where the last inequality follows from (12). ∎

Remark 4.3**.**

Although we will not need it in this generality, we mention that the lemma is true even when only one of the two random variables is assumed to be discrete. Let $X$ be discrete and $Y$ arbitrary, and suppose that \left|\operatorname{corr}\big{(}f(X),g(Y)\big{)}\right|\leq\alpha for any $f$ and any measurable $g$. Then it still follows that $H(X)-H(X|Y)\leq(m-1)\alpha^{2}$.

The point is that one can use (5) to define the conditional entropy $H(X|Y)$ even when $Y$ is not discrete: for an event $A$ the mapping $y\mapsto\mathbb{P}(A|Y=y)$ needs to be replaced by the conditional expectation $\mathbb{E}(\mathbbm{1}_{A}|Y)$, which is a measurable function of $Y$. The same modification needs to be made in the above proof.

Now we have all the ingredients to prove Theorem 2.

Proof of Theorem 2.

Let $M$ be finite and let $X_{v}$, $v\in V(T_{d})$, be a factor of i.i.d. process on $M^{V(T_{d})}$. Suppose that the distance of the vertices $u$ and $v$ is $k$ and set

[TABLE]

Then by (9) we know that $\left|\operatorname{corr}(f(X_{u}),f(X_{v}))\right|\leq\alpha$ for any function $f\colon M\to\mathbb{R}$. There is an automorphism of $T_{d}$ taking $u$ to $v$ and $v$ to $u$, which means that the random variables $X_{u}$, $X_{v}$ are exchangeable. Therefore we can apply Lemma 4.1 to $X_{u}$ and $X_{v}$ and we obtain that $\left|\operatorname{corr}(f(X_{u}),g(X_{v}))\right|\leq\alpha$ for any functions $f,g\colon M\to\mathbb{R}$. By Lemma 4.2 it follows that

[TABLE]

and this is exactly what we wanted to prove. ∎

5. Examples

In this section we construct factor of i.i.d. processes showing that our bounds are (essentially) sharp.

5.1. Sharpness of Theorem 1

Let $k$ be a fixed positive integer and $u,v\in V(T_{d})$ vertices at distance $k$. We claim that there exist factor of i.i.d. processes $X$ on $T_{d}$ such that the normalized mutual information $I(X_{u};X_{v})/H(X_{v})$ can be arbitrarily close to the upper bound

[TABLE]

The idea is the following: given i.i.d. labels at each vertex, let the factor process list all the labels within some large distance $R$ at any given vertex. When we look at the joint distribution of $X_{u}$ and $X_{v}$ we get a collection of i.i.d. labels with some labels listed twice. Hence the normalized mutual information is $|B_{R}(u)\cap B_{R}(v)|/|B_{R}(v)|$, where $B_{R}(v)$ denotes the ball of radius $R$ around $v$. It is easy to see that this converges to $\beta_{k}$ as $R\to\infty$.

For a rigorous argument we need to be more careful since listing the labels should be done in an $\operatorname{Aut}(T_{d})$-invariant way. We first introduce two auxiliary lemmas and then precisely define our example.

Lemma 5.1**.**

For any positive integer $L$ there exists a factor of i.i.d. [math]-$1$ labeling of the vertices of $T_{d}$ such that any ball of radius $L$ contains a vertex with label $1$ but any two vertices of label $1$ have distance greater than $L$.

Lemma 5.2**.**

For any positive integer $L$ there exists a factor of i.i.d. coloring of the vertices of $T_{d}$ such that finitely many colors are used and vertices of the same color have distance greater than $L$.

Example 5.3**.**

Given $k$ and $R$, let $C=\left(C_{w}\right)_{w\in V(T_{d})}$ be a factor of i.i.d. coloring provided by Lemma 5.2 for $L=2R+k$. For a positive integer $N$ let $Z_{w}$, $w\in V(T_{d})$, be i.i.d. uniform labels on $\{1,2,\ldots,N\}$. We set

[TABLE]

Then for vertices $u,v$ at distance $k$ we have

[TABLE]

Indeed, $X_{v}$ can be viewed as the list of variables $(C_{w},Z_{w})$, $w\in B_{R}(v)$, ordered by $C_{w}$ (which are all different). This clearly defines an $\operatorname{Aut}(T_{d})$-factor of i.i.d. process $X_{v}$, $v\in V(T_{d})$. Conditioned on the coloring process $C$, the entropies are easy to compute:

[TABLE]

Since the contribution of the coloring to the entropies does not depend on $N$, it gets negligible when $N$ is large enough, and (14) follows.

Finally, we prove the two lemmas.

Proof of Lemma 5.1.

We describe the labeling as the output of a randomized local algorithm, which is easy to interpret as a factor of i.i.d. process.

In the beginning all labels are undefined. The algorithm consists of countably many steps. At every odd step every vertex with undefined label proposes to get a label $1$ with probability $1/2$. Suppose that a vertex $v$ proposes to get label $1$. If there is no other vertex within distance $L$ of $v$ that also proposes to get label $1$, then the label of $v$ is fixed, otherwise the proposed label is withdrawn. At even steps, undefined vertices check if a label $1$ has appeared within distance $L$ and set their own label [math] if this is the case.

Note that at an odd step any undefined label gets fixed with probability greater than some positive constant $\varepsilon$ depending on $L$. It follows that after countably many steps all labels will be defined with probability $1$. It is easy to verify that the obtained labeling has all the required properties. ∎

Proof of Lemma 5.2.

Lemma 5.1 is used to find vertices with color $1$. A similar algorithm is applied for color $2$, but now some vertices already have defined labels when launching the algorithm. We continue by adding more colors the same way.

After having added $n$ colors this way, every ball of radius $L$ around an uncolored vertex must contain vertices of each color $1,2,\ldots,n$. When $n$ becomes equal to the number of vertices in a ball of radius $L$, this is not possible any longer, therefore we cannot have any more uncolored vertices at that point, meaning that we have colored all vertices in the required manner using at most $n$ colors. ∎

5.2. Sharpness of Theorem 2

The next example shows that the bound obtained in Theorem 2 is essentially sharp. First we briefly describe the construction. We start with an i.i.d. process where each label has standard normal distribution. Then we take a linear factor: each new label is some linear combination of the i.i.d. labels. We will choose the coefficients in a way that the correlation decay for the obtained factor process is close to the bound (9). Then we take the sign of the label of this factor process at every vertex. We will see that for this $\{\pm 1\}$-valued process the correlation decays at roughly the same rate. However, for symmetric binary variables the mutual information is essentially the square of the correlation.

More precisely, for any $\varepsilon>0$ we construct a factor of i.i.d. process (with two states) such that the mutual information for distance $k$ is $\Omega\left(k^{2-\varepsilon}(d-1)^{-k}\right)$.

Example 5.4**.**

Fix a parameter $\varepsilon>0$. Let $Z_{w}$, $w\in V(T_{d})$, be i.i.d. standard normal random variables. We first define a factor $Y$ of the i.i.d. process $Z$ by taking linear combinations of $Z_{w}$ with the following coefficients:

[TABLE]

Then apply the sign function at each vertex:

[TABLE]

Note that $Y_{v}$ is well defined since the sum of the squares of the coefficients is finite. Therefore $Y_{v}$ is a normal random variable with mean [math] and some positive and finite variance $\gamma=\gamma(\varepsilon)$. From this point on $\gamma$ will denote a positive constant that depends only on $\varepsilon$ (possibly a different constant at each occurrence).

Suppose that $u$ and $v$ have distance $k$. We denote the unique path connecting them by $u_{0}=u,u_{1},\ldots,u_{k-1},u_{k}=v$. If we are at vertex $u_{j}$, $1\leq j\leq k-1$, and move distance $n$ away from the path, then we get to a vertex $w$ for which $\operatorname{dist}(u,w)=j+n$ and $\operatorname{dist}(v,w)=k-j+n$. The number of such vertices is clearly $(d-2)(d-1)^{n-1}$. Thus

[TABLE]

We ignore the terms for which $j+n<k$ and rearrange the rest of the sum grouping the terms based on the value $m\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=j+n$. For a given $m\geq k$ and $j\in\{1,\ldots,k-1\}$ we have $n=m-j$ and hence $k-j+n=k+m-2j$. Therefore the average of $k-j+n$ for a given $m$ as $j$ runs through $1,\ldots,k-1$ is exactly $m$, and consequently the convexity of $x^{-\frac{1}{2}-\varepsilon}$ implies that

[TABLE]

It follows that

[TABLE]

and the same is true for $\operatorname{corr}(Y_{u},Y_{v})$ (again with a different $\gamma$). Note that there exist constants $0<\gamma_{1}<\gamma_{2}$ such that for any $W,W^{\prime}$ jointly normal random variables we have

[TABLE]

This means that we get the same correlation (up to a constant factor) after taking the sign of $Y$:

[TABLE]

Now working with symmetric binary variables, elementary computations show that when $P(X_{u}=X_{v})$ is close to $1/2$, we have

[TABLE]

and

[TABLE]

for some constants $0<\gamma_{1}<\gamma_{2}$. It follows that

[TABLE]

which indeed confirms that the bound in Theorem 2 is essentially sharp.