Percolation V: Kesten’s Theorem

Last time we outlined the four steps of the proof of Harris-Kesten and worked through the first. This time, we’ll do the second: showing that for {p>1/2}, there is an infinite open cluster with probability 1, allowing us to conclude that {p_H=\frac{1}{2}}. We begin with two lemmas.

1. Some Lemmas

Recalling our discussion of the combinatorial hypercube, we say that a bond {e} is pivotal for an event {E} in a configuration {\omega} if switching the state of {e} takes {\omega} into or out of {E}. The influence of {e} on {E} is

\displaystyle I_p(e,E)=\mathop{\mathbb P}_p(e\text{ is pivotal for }E).

For {E} increasing, this simplifies to

\displaystyle \mathop{\mathbb P}_p(\omega^+\in E, \omega^{-}\not\in E),

where {\omega^{\pm}} are configurations that differ from {\omega} only at {e}.

Lemma 1. Let {R} be an {m\times n} rectangle in {{\mathbb Z}^2}. For a bond {e} in {R},

\displaystyle I_p(e, H(R))\leq 2\mathop{\mathbb P}_{1/2}(r(C_0)\geq \min\{m/2-1, (n-1)/2\})

for {0<p<1}.

Proof: For this proof, we ignore all bonds outside {R}. The argument is a little weird: we’ll prove that

\displaystyle I_p(e, H(R))\leq 2\mathop{\mathbb P}_p(r(C_0)\geq m/2-1)

and

\displaystyle O_p(e, H(R))\leq 2\mathop{\mathbb P}_{1-p}(r(C_0)\geq (n-1)/2).

Then, noting that {\mathop{\mathbb P}_p(r(C_0)\geq a)} is an increasing function of {p}, the first allows us to conclude for {p\leq 1/2} and the second for {p\geq 1/2}.

Since {\omega^+\in H(R)} and {\omega^-\not\in H(R)}, any horizontal crossing of {R} uses {e}. Thus, in {\omega}, each endpoint of {e} is joined by an open path to either end of {R}, at least one of which must have length {\geq m/2-1}, so the first bound given above follows.

Since {\omega^-\not \in H(R)}, a previous lemma tells us that {\omega^-\in V(R^h)}. Similarly, {\omega^+\not \in V(R^h)}, so {\omega^-} contains an open dual path crossing {R^h} and using the dual edge of {e}. So that edge has an endpoint which is the start of an open dual path of length {(n-1)/2}, and so we get the second bound stated above. \Box

Lemma 2. Fix {p>1/2} and an integer {\rho>1}. There exist constants {\gamma} (that depends on {\rho}) and {n_0} (that depends on {p} and {\rho}) such that

\displaystyle h_p(\rho n,n)\geq 1-n^{-\gamma}

for all {n\geq n_0}.

Proof: Recall that in a remark last time, we mentioned that

\displaystyle h(m_1+m_2-2n, 2n)\geq \frac{h(m_1, 2n)h(m_2, 2n)}{2}.

Setting {m_1=m, m_2=3n} gives

\displaystyle h(m+n, 2n)\geq \frac{h(m,2n)h(3n,2n)}{2}\geq \frac{h(m,2n)}{2^8},

and it follows from this that

\displaystyle h(kn,2n)\geq 2^{17-8k}

for {k\geq 3}. (We’re using some bounds on {h(3n, 2n)} from last post here.) Since {h(m, 2n+1)\geq h(m, 2n)}, there must be a constant {h_k} such that {h(kn,n)\geq h_k}. The consequence of this we need here is that {\mathop{\mathbb P}_{1/2}(H(R))\geq h_{\rho}.}

Now, in the proof of Harris’s theorem, we showed that

\displaystyle \mathop{\mathbb P}_{1/2}(r(C_0)\geq n)\leq n^{-c},

and combining this with the previous lemma gives

\displaystyle I_{p'}(e, H(R))\leq n^{-a}=\delta

for all bonds {e\in R} and {p'\in [0.5,p]}. Let {f(p')=\mathop{\mathbb P}_{p'}(H(R))}. Using a lemma proved in the second post in this series, we get the bound

\displaystyle \sum_{e\in H(R)} I_{p'}(e, H(R))\geq cf(p')(1-f(p'))\log(1/\delta).

But another result from that second post tells us that the sum on the LHS is the derivative of {f(p')}, so we let

\displaystyle g(p')=\log\frac{f(p')}{1-f(p')},

and write

\displaystyle \frac{d}{dp'}g(p')=\frac{1}{f(p')(1-f(p'))}\frac{d}{dp'}f(p')\geq c\log(1/\delta)=ac\log n.

Our observation above that {\mathop{\mathbb P}_{1/2}(H(R))\geq h_{\rho}} says that {g(1/2)} is bounded below by a constant depending on {\rho}. So for {n\geq n_0}, we get {g(p)\geq ac(p-0.5)\log n}, and the result follows. \Box

2. Kesten’s Theorem

This is the last piece in the proof that {p_H({\mathbb Z}^2)=\frac{1}{2}}.

Theorem 3 (Kesten’s Theorem). Let {E_{\infty}} be the event that there is an infinite open cluster. Then for bond percolation in {{\mathbb Z}^2}, if {p>1/2}, we have {\mathop{\mathbb P}_p(E_{\infty})=1.}

Proof: Take {p>1/2} and choose {\gamma} (depending on {p}) and {n_0} be as in the second lemma with {\rho=2}. Choose some large {n}, and let {R_1,\cdots} be a family of rectangles whose bottom left corner is the origin and which have sides of length {2^kn} and {2^{k+1}n}. The longer side is the vertical for even {k} and the horizontal for odd {k}.

Suppose {E_k} is the event that {R_k} is crossed in the longer direction by an open path. Any two such paths must meet, so if every {E_k} holds, so does {E_{\infty}} (since the open cluster containing the origin has unbounded radius). Now we write

\displaystyle \mathop{\mathbb P}_p(E_{\infty})\geq \mathop{\mathbb P}_p\left(\bigcap_{k=0}^{\infty} E_k\right)=1-\mathop{\mathbb P}_p\left(\bigcup_{k=0}^{\infty} E_k^c\right)\geq 1-\sum_{k=0}^{\infty} \mathop{\mathbb P}_p(E_k^c).

But by the final lemma of the previous section, and the fact that {n} is arbitrarily large,

\displaystyle \sum_{k=0}^{\infty} \mathop{\mathbb P}(E_k^c)\leq \sum_{k=0}^{\infty} (2^kn)^{-\gamma}=\frac{n^{-\gamma}}{1-2^{-\gamma}}<1.

So {\mathop{\mathbb P}_p(E_{\infty})>0}, and thus 1. \Box

The result proven at the end of the last post was that {p_H>\frac{1}{2}} while this result shows that {p_H\leq \frac{1}{2}}, so we conclude that in fact {p_H=\frac{1}{2}}. In the next and final post, we’ll turn our attention to {p_T}.

Percolation IV: Harris’s Theorem

We’re ready to start proving the Harris-Kesten theorem, so from now on, everything we say is about bond percolation in {{\mathbb Z}^2}. The global structure of the argument is in three steps:

  1. Show that {\theta(1/2)=0} (this implies {p_H\geq 1/2})
  2. Show that for {p>1/2}, there is an infinite open cluster with probability 1 (this implies {p_H\leq 1/2})
  3. Show that {\chi(p)<\infty} for {p<1/2} (this implies that {p_T\geq 1/2})
  4. Conclude that {p_T=p_H=1/2}

This post will accomplish the first step.

1. Crossing Probabilities

We start with some more definitions: a rectangle {R} in {{\mathbb Z}^2} is a set of sites of the form {[a,b]\times [c,d]}. We say this rectangle has size {k\times \ell}, where {k=b-a+1, \ell=d-c+1}. It contains {k\ell} sites and {2k\ell-k-\ell} bonds.

This rectangle has two duals: a horizontal dual

\displaystyle R^h=[a+0.5, b-0.5]\times [c-0.5, d+0.5]

and a vertical dual

\displaystyle R^v=[a-0.5, b+0.5]\times [c+0.5, d-0.5].

We have that {R^h} is a {k-1\times \ell+1} rectangle and {R^v} is a {k+1\times \ell-1} rectangle. Additionally,

\displaystyle \left(R^h\right)^v=\left(R^v\right)^h=R.

Finally, we define an open horizontal crossing of a rectangle {R} to be an open path in {R} connecting a site {(a,x)} to a site {(b,y)}, and similarly for an open vertical crossing. We write {H(R)} and {V(R)} for the events that such crossings exist.

The following lemma is, according to Bollobas, the “reason” why {p_H=1/2}. It is intuitively clear, but working through all the details is a bit of slog, so let’s just take it at face value.

Lemma 1. Let {R} be a rectangle in {{\mathbb Z}^2} or its dual. Then exactly one of {H(R)} and {V(R^h)} holds.

Why exactly is this lemma the “reason” cited above? The following corollary sheds some light on that.

Corollary 2. If {R} and {R'} are {k\times \ell-1} and {k-1\times \ell} rectangles in {{\mathbb Z}^2}, then

\displaystyle \mathop{\mathbb P}_p(H(R))+\mathop{\mathbb P}_{1-p}(V(R'))=1.

Proof: The bonds of the dual are open when the bonds of the original graph are closed and vice versa. So by Lemma 1, each configuration of bonds will satisfy exactly one of {H(R)} and {V(R^h)}, so

\displaystyle \mathop{\mathbb P}(H(R))+\mathop{\mathbb P}(V(R^h))=1.

But {R^h} is indistinguishable from {R'}, so the result follows. \Box

Now if we take {k=\ell} in this corollary, we get that for an {n+1\times n} rectangle {R},

\displaystyle \mathop{\mathbb P}_{1/2}(H(R))+\mathop{\mathbb P}_{1/2}(V(R'))=1.

But because the rectangle {R'} has dimension {n\times n+1}, the event {V(R')} has the same probability as {H(R')}, so they are both equal to {1/2}.

So the intuition is somehow that when {p>1/2}, it’s more likely you have a horizontal crossing than not, so you get an infinite open cluster. When {p<1/2}, it’s less likely than not, so you don’t. The critical behavior is at exactly {p=1/2}.

More importantly, we get that if {S} is an {n\times n} square,

\displaystyle \mathop{\mathbb P}_{1/2}(V(S))=\mathop{\mathbb P}_{1/2}(H(S))\geq \frac{1}{2},

where we have the inequality because this event must be more likely than the crossing for an {n\times n+1} rectangle. This is a bound on the crossing probabilities of a square. We now have to extend this result to rectangles.

2. Squares to Rectangles

This next lemma relates crossings of squares to crossings of rectangles. We use the notation {[n]=[0,n]}.

Lemma 3. Consider {R=[m]\times [2n]}, an {m\times 2n} rectangle with {m\geq n}. Let {X(R)} be the event that there are open paths {P_1, P_2} such that {P_1} crosses the {n\times n} square {S=[n]\times [n]} from top to bottom and {P_2} lies within {R} and joins some site on {P_1} to the right hand side of {R}. Then

\displaystyle \mathop{\mathbb P}_p(X(R))\geq \frac{\mathop{\mathbb P}_p(H(R))\mathop{\mathbb P}_p(V(S))}{2}.

Proof: Suppose {V(S)} occurs, so an open path {P_0} crosses {S} from top to bottom. Such a path cuts {S} into two pieces, one on the left and on the right. Let {LV(S)} be the left-most open vertical crossing (when one exists). For any {P_1}, the event {\{LV(S)=P_1\}} does not depend on bonds to the right of {P_1}.

The key claim is that

\displaystyle \mathop{\mathbb P}_p(X(R)\mid LV(S)=P_1)\geq \frac{\mathop{\mathbb P}(H(R))}{2}.

Since {V(S)} is a disjoint union of events of the form {\{LV(S)=P_1\}}, this implies that

\displaystyle \mathop{\mathbb P}_p(X(R)\mid V(S))\geq\frac{\mathop{\mathbb P}_p(H(R))}{2},

which rearranges to

\displaystyle \mathop{\mathbb P}_p(X(R)\cap V(S))\geq \frac{\mathop{\mathbb P}_p(H(R))\mathop{\mathbb P}_p(V(S))}{2}.

But {X(R)} occurring means that {V(S)} does as well, so {\mathop{\mathbb P}_p(X(R)\cap V(S))=\mathop{\mathbb P}_p(X(R))}, and the lemma is proven.

For the claim, let {P} be the path (that may or may not be open) formed by the union of {P_1} and it’s reflection {P_1'} in the horizontal axis of {R}, with a bond joining {P_1} to {P_1'}. This path is a vertical crossing of the rectangle {[n]\times [2n]}.

Now, there is an open path {P_3} crossing {R} horizontally with probability {\mathop{\mathbb P}_p(H(R))}. This path must meet {P}, and the probability it does so at a site of {P_1} is at least {\mathop{\mathbb P}_p(H(R))/2}. Thus the event that there is an open path {P_2} in {R} to the right of {P} joining a site of {P_1} to the right end of {R} has probability at least {\mathop{\mathbb P}_p(H(R))/2}. This event depends only bonds to the right of {P}, and all such bonds in {S} are to the right of {P_1}. These bonds are independent of {LV(S)=P_1}, so

\displaystyle \mathop{\mathbb P}_p(Y(P_1)\mid LV(S)=P_1)=\mathop{\mathbb P}_p(Y(P_1))\geq \mathop{\mathbb P}_p(H(R))/2.

If {Y(P_1)} and {LV(S)=P_1} both occur, then {X(R)} does as well. So

\displaystyle \mathop{\mathbb P}_p(X(R)\mid LV(S)=P_1)\geq \mathop{\mathbb P}_p(H(R))/2,

proving the claim and thus the lemma. \Box

Remark 1. If we instead take an {m_!\times 2n} rectangle {R} and an {m_2\times 2n} rectangle {R'}, this proof will show that

\displaystyle h(m_1+m_2-n, 2n)\geq \frac{h(m_1, 2n)h(m_2,2n)}{2^5},

and we will use this formulation later on.

(This argument is much simpler than it seems – drawing the picture goes a long way.)

3. Thinning the Rectangles Out

With this lemma, we can now bound the crossing probabilities for non-square rectangles. In particular, we look at a {3n\times 2n} rectangle. For convenience, let {h_p(m,n)=\mathop{\mathbb P}_p(H(R))}, and since it’s the case we really care about, {h(m,n)=h_{1/2}(m,n)}.

Corollary 4. We have {h(3n, 2n)\geq 2^{-7}}.

Proof: Take two square {2n\times 2n} rectangles {R, R'} that overlap in an {n\times 2n} strip. This {n\times 2n} strip contains two {n\times n} squares; call {S} the smaller of these. (Again, draw the picture!) Let {X'(R')} be the event {X(R)} described above, but reflected horizontally. Then we apply Lemma 4 to {R} (which is coincidentally a square) to get

\displaystyle \mathop{\mathbb P}_p(X'(R'))=\mathop{\mathbb P}_p(X(R))\geq \frac{\mathop{\mathbb P}_p(H(R))\mathop{\mathbb P}_p(V(S))}{2}.

Now, we call back to Harris’s lemma from the previous post: the events {X(R), X'(R'), H(S)} are all increasingly and positive correlated. If they all hold, so does {H(R\cup R')}, and so we have

\displaystyle h(3n,2n)=\mathop{\mathbb P}_{1/2}(H(R\cup R'))\geq \mathop{\mathbb P}_{1/2}(X'(R'))\mathop{\mathbb P}_{1/2}(X(R))\mathop{\mathbb P}_{1/2}(H(S)).

This is then

\displaystyle \geq \frac{\mathop{\mathbb P}_{1/2}(H(R))^2\mathop{\mathbb P}_{1/2}(V(S))^2\mathop{\mathbb P}_{1/2}(H(S))}{4}.

But by our observation at the end of Section 1, this is at least

\displaystyle \frac{(0.5^2)^3}{4}=2^{-7},

as desired. \Box

A {3n\times 2n} rectangle isn’t quite thing enough for our purposes, but the rest of the thinning can be done using the remark after Lemma 3. We have

\displaystyle h(5n,2n)\geq \frac{h(3m,2n)^2}{2^5}\geq 2^{-19},

and so

\displaystyle h(6n,2n)\geq \frac{h(5n,2n)h(2n,2n)}{2^5}\geq 2^{-25},

and this is the bound that our proof of {\theta(1/2)=0} will hinge on.

4. Harris’s Theorem

Theorem 5. For bond percolation on {{\mathbb Z}^2}, {\theta(1/2)=0}.

Proof: Let

\displaystyle r(C_0)=\sup\{d(x,0)\mid x\in C_0\}.

The actual result we’ll show is that

\displaystyle \mathop{\mathbb P}_{1/2}(r(C_0)\geq n)\leq n^{-c}

for a constant {c}, and the stated result follows from this.

When {p=1/2}, bonds in the dual lattice are open with probability {1/2}, so by the computation above, the probability a {6n\times 2n} rectangle has a crossing the long way is at least {2^{-25}}. Consider 4 such rectangles oriented to form an annulus around the origin, with the “hole” in the annulus being a {2n\times 2n} square centerd at the origin.

By Harris’s lemma, the probability each of these rectangles is crossed the long way by an open path in the dual is at least {\epsilon=2^{-100}}. The union of these paths is an open dual cycle around the annulus.

Let {A_k} be the square annulus centered on {(0.5, 0.5)} with inner radius {4^k} and outer radius {3\cdot 4^k}. Let {E_k} be the event that {A_k} contains an open dual cycle surrounding the inside of {A_k}, and thus the origin.

By our argument above, {\mathop{\mathbb P}(E_k)\geq \epsilon} for each {k}. All of the {E_k} are independent, and if {E_k} holds, then there is no path from inside {A_k} to outside {A_k}, so {r(C_0)\leq 1+3\cdot 4^k\leq 4^{k+1}.}

Then,

\displaystyle \mathop{\mathbb P}\left(r(C_0)\geq 4^{\ell+1}\right)\leq \mathop{\mathbb P}_{1/2}\left(\bigcap_{k=1}^{\ell} E_k^c\right)=\prod_{k=1}^{\ell} \mathop{\mathbb P}_{1/2}(E_k^c)\leq (1-\epsilon)^{\ell},

which is the desired bound. \Box

Percolation III: Some Lemmas

The next chapter of Bollobas is a collection of probabilistic tools used throughout the book; this post presents two of these results that will be key to our proof of the Harris-Kesten theorem. Both of these results are set in the framework of weighted hypercubes, which we now introduce.

Take a set {S} and its power set {\mathcal{P}(S)}. This power set can be identified with the set of functions {f: S\rightarrow \{0,1\}} by identifying {A\in \mathcal{P}(S)} with its characteristic function. In the case where {S} is finite (and this is the case we care about), we can think of {S} as {\{1,\cdots, n\}} and {\mathcal{P}(S)=\mathcal{P}(n)} as the cube {Q^n=\{0,1\}^n.} This cube is the set of binary sequences of length {n}, where a subset {A\subset S} is associated to a sequence that has a 1 in position {i} if {i\in A} and a 0 otherwise.

More geometrically, we can think of {Q^n} as a graph whose vertices are {\mathcal{P}(n)}. Two sets {A, B} are connected in this graph if {|A\Delta B|=1} (i.e. {A} and {B} differ at one entry in their sequences). Put yet another way, if {\mathbf{a}, \mathbf{b}\in \{0,1\}^n} are two vertices, they are connected if {a-b=\pm \mathbf{e}_i}, where {\mathbf{e}_i} is a standard basis vector. We can make {Q^n} a directed graph by saying the edge {\mathbf{a}\mathbf{b}} is oriented from {\mathbf{a}} to {\mathbf{b}} if {\mathbf{b}-\mathbf{a}=\mathbf{e}_i}. We write this as {\mathbf{a}\leq \mathbf{b}}.

Now, shifting gears, the points of {Q^n} are associated with the outcome of a sequence of {n} coin tosses. If {\mathbf{p}=(p_1,\cdots, p_n)} is the sequence of probabilities of success for these {n} coin tosses, then { \mathbf{p}} induces a natural probability measure on {Q^n}.

For example, if {\mathbf{p}=(0.5,\cdots, 0.5)} we obtain the uniform measure, and if {\mathbf{p}=(p,\cdots, p)}, we obtain {\mathop{\mathbb P}(A)=p^{|A|}(1-p)^{n-|A|}}. We write {Q_p^n} for the cube with this measure, and it is equivalent to the standard percolation models.

We write {Q_{\mathbf{p}}^n} for the probability space {(\mathop{\mathbb P}_{\mathbf{p}}, Q^n)} and distinguish two kinds of events. A subset {U\subset Q^n} is an up-set if {a,b\in Q^n, a\leq b\implies b\in U}. Similarly, {U} is a down-set if {a,b \in Q^n, a\geq b\implies b\in U}. (Keep in mind that {U} is a subset of {Q^n}, which is identified with {\mathcal{P}(S)}, so {a, b} are subsets of the original set {S}.) The corresponding events are referred to as increasing and decreasing respectively.

With that set up out of the way, we can state our first lemma.

1. Harris’s Lemma

This lemma is the intuitive fact that increasing events are positively correlated. Let’s do a bit more set up.

For a set {A\subset Q^n} and {t=0,1} we can define

\displaystyle A_t=\{(a_i)_{i=1}^{n-1}\mid (a_1,\cdots, a_{n-1}, t)\in A\}\subset Q^{n-1}.

If {A} is an up-set, {A_0\subset A_1}, and if {A} is a down-set, {A_1\subset A_0} (because {A_1, A_0} always differ at only one position, so their difference is always a basis vector, so it’s just a matter of sign).

If we write {\mathop{\mathbb P}} for the probability measure on both {Q_{(p_1,\cdots, p_n)}^n} and {Q_{(p_1,\cdots, p_{n-1})}^{n-1}}, we have

\displaystyle \mathop{\mathbb P}(A)=(1-p_n)\mathop{\mathbb P}(A_0)+p_n\mathop{\mathbb P}(A_1).

Finally, the result:

Lemma 1. Let {A, B} be subsets of {Q_{\mathbf{p}}^n}. If they are both up-sets or both down-sets, then {\mathop{\mathbb P}(A\cap B)\geq \mathop{\mathbb P}(A)\mathop{\mathbb P}(B)}. If {A} is an up-set and {B} is a down-set, then {\mathop{\mathbb P}(A\cap B)\leq \mathop{\mathbb P}(A)\mathop{\mathbb P}(B)}.

(This implies the correlation statement, since the first condition rearranges to {\mathop{\mathbb P}(A\mid B)\geq \mathop{\mathbb P}(A)} and similarly for the second.)

Proof: We induct on {n}. The case {n=1} is trivial, so suppose the result holds for {n-1}.

If {A} and {B} are both up-sets or both down-sets, then either {A_0\subset A_1} and {B_0\subset B_1} or {A_1\subset A_0} and {B_1\subset B_0}. This means that

\displaystyle (\mathop{\mathbb P}(A_0)-\mathop{\mathbb P}(A_1))(\mathop{\mathbb P}(B_0)-\mathop{\mathbb P}(B_1))\geq 0.

Our argument will rely on the equivalent fact that

\displaystyle \mathop{\mathbb P}(A_0)\mathop{\mathbb P}(A_1)+\mathop{\mathbb P}(B_0)\mathop{\mathbb P}(B_1)\geq \mathop{\mathbb P}(A_1)\mathop{\mathbb P}(B_0)+\mathop{\mathbb P}(A_0)\mathop{\mathbb P}(B_1).

Now we write

\displaystyle \mathop{\mathbb P}(A\cap B)=(1-p_n)\mathop{\mathbb P}(A_0\cap B_0)+p_n \mathop{\mathbb P}(A_1\cap B_1)

which is

\displaystyle \geq (1-p_n)\mathop{\mathbb P}(A_0)\mathop{\mathbb P}(B_0)+p_n\mathop{\mathbb P}(A_1)\mathop{\mathbb P}(B_1).

We cleverly rewrite this as

\displaystyle (1-p_n)^2\mathop{\mathbb P}(A_0)\mathop{\mathbb P}(B_0)+p_n^2\mathop{\mathbb P}(A_1)\mathop{\mathbb P}(B_1)+(1-p)p(\mathop{\mathbb P}(A_0)\mathop{\mathbb P}(B_0)+\mathop{\mathbb P}(A_1)\mathop{\mathbb P}(B_1)),

(yes, they’re really equal!) and use our earlier inequality to see that this is

\displaystyle \geq (1-p_n)^2\mathop{\mathbb P}(A_0)\mathop{\mathbb P}(B_0)+p_n^2\mathop{\mathbb P}(A_1)\mathop{\mathbb P}(B_1)+(1-p)p(\mathop{\mathbb P}(A_0)\mathop{\mathbb P}(B_1)+\mathop{\mathbb P}(A_1)\mathop{\mathbb P}(B_0).

But this last line is equal to

\displaystyle \left[(1-p_n)\mathop{\mathbb P}(A_0)+p_n\mathop{\mathbb P}(A_1)\right]\left[(1-p_n)\mathop{\mathbb P}(B_0)+p_n\mathop{\mathbb P}(B_1)\right]=\mathop{\mathbb P}(A)\mathop{\mathbb P}(B),

so we conclude.

From this first piece, the other inequality follows easily: if {A} is an up-set and {B} is a down-set, then {B^c} is an up-set, so

\displaystyle \mathop{\mathbb P}(A\cap B)\leq\mathop{\mathbb P}(A)-\mathop{\mathbb P}(A\cap B^c)\leq \mathop{\mathbb P}(A)-\mathop{\mathbb P}(A)\mathop{\mathbb P}(B^c),

from the previous result. This then is

\displaystyle \mathop{\mathbb P}(A)-\mathop{\mathbb P}(A)(1-\mathop{\mathbb P}(B))=\mathop{\mathbb P}(A)\mathop{\mathbb P}(B),

as desired. \Box

2. The Russo-Margulis Lemma

Take an event {A} in the weighted cube {Q_{\mathbf{p}}^n}. We want to understand how {\mathop{\mathbb P}(A)} varies with {\mathbf{p}}.

For a point {\omega\in Q^n}, we say that entry {i}, {\omega_i}, is pivotal for {A} if only one of {\omega} and {\tilde{\omega}} are in {A}, where {\tilde{\omega}} is {\omega} with the entry at {\omega_i} flipped. Then, we define the influence of entry {i} as

\displaystyle \beta_i(A)=\mathop{\mathbb P}(\omega_i\text{ is pivotal for } A)=\mathop{\mathbb P}(\{\omega\mid \omega_i\text{ is pivotal for }A\}).

Lemma 2. Let {A} be an increasing event in {Q_{\mathbf{p}}^n} where {\mathbf{p}=(p_1,\cdots, p_n)}. Then

\displaystyle \frac{\partial}{\partial p_i} \mathop{\mathbb P}(A)=\beta_i(A)\implies \frac{d}{dp}\mathop{\mathbb P}(A)=\sum_{i=1}^n \beta_i(A).

Proof: Assume without the loss of generality that {i=n}. If we have {\mathbf{x}=(x_1,\cdots, x_k)\in Q^k} for {k\leq n}, write {\mathbf{p}^{\mathbf{x}}} for the probability of {x} in the measure induced by {Q_{\mathbf{p}}^n} on {Q^k}. In particular, if {k=n}, then {\mathbf{p}^{\mathbf{x}}=\mathop{\mathbb P}(\{x\})}.

Also, for {\mathbf{x}\in Q^{n-1}}, let {\mathbf{x}_+} and {\mathbf{x}_-} be the events formed by extending {\mathbf{x}} with a 1 or 0 in the last slot, respectively. We have

\displaystyle \mathbf{p}^{\mathbf{x}_+}+\mathbf{p}^{\mathbf{x}_-}=\mathbf{p}^{\mathbf{x}}

by construction.

Now take an up-set {A}, and let {A_a} be the set of {\mathbf{x}\in Q^{n-1}} such that {\mathbf{x}_+} and {\mathbf{x}_-} are both in {A}, and {A_b} be the same set but such that only {\mathbf{x}_+} is. Then,

\displaystyle \mathop{\mathbb P}(A)=\sum_{\mathbf{x}\in A_a} \left(\mathbf{p}^{\mathbf{x}_+}+\mathbf{p}^{\mathbf{x}_-}\right)+\sum_{\mathbf{x}\in A_b}\mathbf{p}^{\mathbf{x}_+},

which, by our earlier observation, becomes

\displaystyle \sum_{\mathbf{x}\in A_a} \mathbf{p}^{\mathbf{x}}+p_n\sum_{\mathbf{x}\in A_b} \mathbf{p}^{\mathbf{x}}.

It’s clear now that

\displaystyle \frac{\partial}{\partial p_n} \mathop{\mathbb P}(A)= \sum_{\mathbf{x}\in A_b} \mathbf{p}^{\mathbf{x}},

but since {A_b} is exactly the set of {\mathbf{x}} for which {n} is pivotal, this sum is {\beta_n(A)}. \Box

That proof is a notational mess, but the idea is simple: think of an element of {Q^n} as an element of {Q^{n-1}}, plus a bit of information that tells you whether it is pivotal, and then use this decomposition to directly take the derivative.

Of course, this result is only useful if you actually understand things about the {\beta_i(A)}. I won’t prove it, but we have this bound: if {\mathop{\mathbb P}(A)=t} and {\beta_i(A)\leq \delta} for each {i}, then

\displaystyle \sum_{i=1}^n \beta_i(A)\geq ct(1-t)\log(1/\delta).

This bound will be the second ingredient necessary in our proof of Harris-Kesten, which we will finally begin with in the next post.

Percolation II: The Integer Lattice

This post presents a couple of results regarding percolation on {{\mathbb Z}^d} (and especially {{\mathbb Z}^2}), which are valuable mainly because they tell us that percolation on these graphs is non-trivial i.e. none of the critical probabilities is 0 or 1. But on a concrete level, they actually do more: they give explicit bounds on the critical probabilities.

Proposition 1. For bond percolation on {{\mathbb Z}^2}, we have {p_T\geq \frac{1}{3}}

This proof hinges on counting {\mu_n({\mathbb Z}^2)}, the number of self-avoiding walks of length {n} (in the language of graph theory, paths) on {{\mathbb Z}^2}. We will make use of the obvious bound

\displaystyle \mu_n({\mathbb Z}^2)\leq 4\cdot 3^{n-1}.

Proof: The middle inequality was discussed in the previous post, so it suffices to do the first and last ones. Let’s start with the first.

Choose some {p<\frac{1}{3}} and consider the bond percolation on {{\mathbb Z}^2} where every bond is open with probability {p}. Let {C_0} be the cluster containing the origin and note that any site in {C_0} must be connected to the origin by an open path starting at the origin. (It’s a little hard to see, but this is where the self-avoiding condition comes in: if the walk from the origin to the site intersected itself, we could “excise” the loop and then get another path that’s more likely to be open, because it is shorter.) The size of {C_0} is bounded by the number {X} of such paths and the probaibility of a path of length {n} being open is {p^n}, so

\displaystyle \mathop{\mathbb E}(|C_0|)\leq \mathop{\mathbb E}(X)=\sum_{n=0}^{\infty} p^n\mu_n\leq 1+\sum_{n=1}^{\infty} p^n\cdot 4\cdot 3^{n-1}<\infty,

because {p<\frac{1}{3}}. Since the expected size of {C_0} is finite for any {p<\frac{1}{3}}, we conclude that {p_T\geq \frac{1}{3}}. \Box

Proposition 2. For bond percolation on {{\mathbb Z}^2}, we have {p_H\leq \frac{2}{3}}

This proof hinges on studying the dual graph of {{\mathbb Z}^2} (more on this in a second) and using a technical lemma that I won’t prove.

For a graph {\Lambda}, the dual {\Lambda^*} is a graph whose vertices correspond to faces of {\Lambda} with edges connecting adjacent faces in {\Lambda}. For example,

\displaystyle \left({\mathbb Z}^2\right)^*={\mathbb Z}^2+(0.5,0.5).

In the context of percolation, we call the sites and bonds of {\Lambda^*} dual sites and dual bonds, with a dual bond taken to be closed when the original bond is open, or vice versa. (We’re associating bonds to dual bonds here by {e\mapsto e^*} where {e^*} is the edge of {\Lambda^*} connecting the faces of {\Lambda} that meet at {e}).

In {{\mathbb Z}^2}, if {H} is a finite subgraph of {{\mathbb Z}^2} with vertex set {C}, then {{\mathbb Z}^2\setminus C} must have some infinite component {C_{\infty}}. We define {\partial^{\infty} C} to be the bonds of {\left({\mathbb Z}^2\right)^*} that are dual to bonds of {{\mathbb Z}^2} joining {C} and {C_{\infty}}.

Put another way, you take some finite subgraph, and look at all the edges connecting it to the infinite component of it’s complement. This set of edges maps to some set of edges in the dual, and that set is {\partial^{\infty} C}. The lemma says that {\partial^{\infty} C} looks like the boundary of {H}.

Lemma 3. {\partial^{\infty} C} is a cycle containing {C}.

Proof: As before, take some {p>\frac{2}{3}}, and consider bond percolation on {{\mathbb Z}^2} where each bond is open with probability {p}. Take dual bonds {e^*} to be open if {e} is closed and vice versa, and call a cycle in the dual consisting of open bonds an open dual cycle.

Let {L_k} be the segment connecting the origin to {(k,0)} and {S} some dual cycle of length {2\ell} surrounding {L_k}. {S} must cut the {x}-axis between {k+0.5} and {0.5(2\ell-3)}, so there are fewer than {\ell} choices for the dual bond that cuts the {x}-axis. The rest of {S} is a path of length {2\ell-1}, and since the dual of {{\mathbb Z}^2} is isomorphic to {{\mathbb Z}^2}, there are {\mu_{2\ell-1}} possibilities for this portion of {S}. In total, we have fewer than {\ell \mu_{2\ell-1}} possibilities for {S}.

Let {Y_k} be the number of open dual cycles surrounding {L_k}. An open dual cycle of length {2\ell} occurs with probability {(1-p)^{2\ell}}, so

\displaystyle \mathop{\mathbb E}(Y_k)\leq \sum_{\ell=k+2}^{\infty} \ell\mu_{2\ell-1}(1-p)^{2\ell}\leq \sum_{\ell =k+2}^{\infty} \ell\cdot 4\cdot 3^{2\ell-2}(1-p)^{2\ell}<\infty,

because {p>\frac{2}{3}\implies 3(1-p)<1}.

This means that {\mathop{\mathbb E}(Y_k)\rightarrow 0} as {k\rightarrow \infty}, and in particular, {\mathop{\mathbb E}(Y_k)<1} eventually. If {A_k} is the event that {Y_k=0}, this means that {\mathop{\mathbb P}(A_k)>0} (if {P(A_k)} were never zero, then integrating over a set of measure 1 would have to give something that is at least 1). Now let {B_k} be the event that all the bonds in {L_k} are open, which occurs with probability {p^k}, which is independent of {A_k}.

If {A_k} and {B_k} both occur, then {C_0} cannot be contained in a cycle of the dual graph (since {Y_k=0}), so it must be infinite by our lemma. Thus

\displaystyle \theta_0(p)=\mathop{\mathbb P}(|C_0|=\infty)\geq \mathop{\mathbb P}(A_k\cap B_k)= \mathop{\mathbb P}(A_k)\mathop{\mathbb P}(B_k)\geq \mathop{\mathbb P}(A_k)p^k>0,

so there is an infinite open cluster with probability 1. This means that {p_H\geq p}, and since {p} is any probability {>\frac{2}{3}}, we have {p_H\leq \frac{2}{3}}. \Box

Remark 1. \leavevmode

  1. Recalling that {p_T\leq p_H}, we can chain these together into

    \displaystyle \frac{1}{3}\leq p_T\leq p_H\leq \frac{2}{3},

    showing that the critical probabilities for bond percolation on {{\mathbb Z}^2} are non-trivial.

     

  2. The bound of {\mu_n\leq 4\cdot 3^{n-1}} is crude, and these inequalities can be sharpened by using tighter inequalities. For example, noting that {\mu_{m+n}\leq \mu_m\mu_n} (you can just concatenate paths), we see by Fekete’s lemma that {\mu_n^{1/n}\rightarrow \lambda}. This {\lambda} is called the connective constant, and our result becomes

    \displaystyle \frac{1}{\lambda}\leq p_T\leq p_H\leq 1-\frac{1}{\lambda},

    so we just need to estimate this constant, which I won’t get into but can certainly be done.

     

  3. Since {{\mathbb Z}^d} contains {{\mathbb Z}^2} within it for {d\geq 2}, we can recall the result at the end of the previous post to obtain

    \displaystyle \frac{1}{2d-1}\leq p_T({\mathbb Z}^d)\leq p_H({\mathbb Z}^d)\leq \frac{2}{3},

    giving non-triviality of critical probabilities for {{\mathbb Z}^d}.

 

These results are only for bond percolation. We can use them to say things about site percolation as well though, by showing that percolation is “more likely” in the site model than the bond model.

Theorem 4. For any graph {\Lambda}, we have {p_H^{\mathrm{s}}\geq p_H^{\mathrm{b}}} and {p_T^{\mathrm{s}}\geq p_T^{\mathrm{b}}}.

Proof: The key claim is that for any site {x}, integer {n}, and probability {p},

\displaystyle \mathop{\mathbb P}^{\text{s}}(|C_x|\geq n)\leq p \mathop{\mathbb P}^{\text{b}}(|C_x|\geq n).

(Here, {\mathop{\mathbb P}^{\text{b}}} and {\mathop{\mathbb P}^{\text{s}}} are the probability measures induced by bond and site percolation where bonds and sites are open with probability {p}, respectively. Throughout this proof, we use such superscripts to indicate the type of percolation being considered.)

From this claim, we let {n\rightarrow \infty} to obtain {\theta_x^{\text{s}}\leq p\theta_x^{\text{b}}}. Then, if {\theta_x^{\text{b}}=0}, we have {\theta_x^{\text{s}}=0} as well, so {p_H^{\text{s}}\geq p_H^{\text{b}}}.

For {p_T}, we argue similarly:

\displaystyle \chi_x^{\text{s}}=\sum_{n=1}^{\infty} \mathop{\mathbb P}^{\text{s}}(|C_x|\geq n)\leq \sum_{n=1}^{\infty} \mathop{\mathbb P}^{\text{b}}(|C_x|\geq n)\leq p\chi_x^{\text{b}},

and so {\chi_x^{\text{b}}<\infty\implies \chi_x^{\text{s}}<\infty}, meaning that {p_T^{\text{s}}\geq p_T^{\text{b}}.}

It remains to show the claim. We begin by writing {\mathop{\mathbb P}^{\text{s}}(|C_x|\geq n)} as

\displaystyle \mathop{\mathbb P}^{\text{s}}(|C_x|\geq n\mid x\text{ is open})\mathop{\mathbb P}^{\text{s}}(x\text{ is open})+\mathop{\mathbb P}^{\text{s}}(|C_x|\geq n\mid x\text{ is closed})\mathop{\mathbb P}^{\text{s}}(x\text{ is closed}).

If {x} is closed, {C_x} is empty, so the second term is zero. Thus we see that

\displaystyle \mathop{\mathbb P}^{\text{s}}(|C_x|\geq n)=p\mathop{\mathbb P}^{\text{s}}(|C_x|\geq n\mid x\text{ is open}),

and so our claim is equivalent to

\displaystyle p\mathop{\mathbb P}^{\text{s}}(|C_x|\geq n\mid x\text{ is open})\geq \mathop{\mathbb P}^{\text{b}}(|C_x|\geq n).

Next, we note that since the event {|C_x^{\text{s}}|\geq n} depends only on vertices that are within {n} of {x}, it suffices to consider the subgraph {\Lambda_n} of points within {n} of {x}. The idea now is to construct a sequence of subsets of {V(\Lambda_n)} (the vertex set of {\Lambda_n)} denoted {\mathcal{T}=(R_t, D_t, U_t)_{t=1}^{\ell}} such that {R_{\ell}= C_x^{\text{s}}}. The constant {\ell} is random, and the letters {R, D, U} stand for reached, dead, and untested respectively.

The construction is as follows:

  1. Let {R_1=\{x\}, U_1=V(\Lambda)\setminus \{x\}, D_1=\emptyset}
  2. Pick a bond {e_t=y_tz_t} where {y_t\in R_t} and {z_t\in U_t}. Remove {z_t} from {U_t}. If {y_t} is open, add it to {R_t} and leave {D_t} alone; otherwise add it to {D_t} and leave {R_t} alone. This gives the new triplet.
  3. Repeat 2 until no such bond {e_t} exists, and set {t=\ell}, completing the process. (This must happen eventually because {\Lambda_n} is finite.)

In the end, {R_{\ell}} is a connected set of open sites with no neighbors in {U_{\ell}} that is disjoint from the sites of {D_{\ell}} which are all closed, so it is exactly {C_x^{\text{s}}}. Note that the conditional probability of {z_t} being open is always {p}.

Now, consider an analogous sequence for {C_x^{\text{b}}}, {\mathcal{T}'}. The difference here is that after picking {e_t}, we check whether {e_t} is open (conditioned on the sequence up to that open). This probability is always {p} as well, so {\mathcal{T}} and {\mathcal{T}'} have the same distribution.

In particular, {|C_x^{\text{s}}|=|R_{\ell}|} and {|R'_{\ell'}|} have the same distribution. Since {R'_{\ell'}} is contained in {C_{x}^{\text{b}}} (it is the vertex set of the subgraph spanned by the bond we found to be open), the claim follows, concluding the proof. \Box

So this shows that {p_H^{\text{s}}, p_T^{\text{s}}} are not zero. To show that they are also not 1, we need another inequality, namely:

Theorem 5. For percolation on an infinite connected graph with maximum degree {\Delta},

\displaystyle p_H^{\mathrm{s}}\leq 1-(1-p_H^{\mathrm{b}})^{\Delta-1}.

The proof is similar, but more involved, so I’ll skip it here. The upshot of all this is that {p_T^{\text{s}}, p_T^{\text{b}}, p_H^{\text{b}}, p_H^{\text{s}}} are all non-trivial for {{\mathbb Z}^2}. In the next few posts, we’ll take the next step and actually determine {p_T^{\text{b}}} and {p_H^{\text{s}}}.

Percolation I: Critical Probabilities

This is the first in a series of posts about percolation theory, which I’m studying this summer. The main text I’m using is Bollobas’s book on the subject, and I plan to follow it pretty closely. There should be around six posts, culminating in the proof of the Harris-Kesten theorem, a major result about percolation on {{\mathbb Z}^2}. This material corresponds to the first three chapters of Bollobas, with some omissions and rearrangement.

1. The Big Picture

The idea is to consider some graph (prototypically {{\mathbb Z}^d} but theoretically any graph), and choose a random subgraph of this graph by selecting each edge to be part of the subgraph with probability {p}. Alternatively, do the same thing with vertices instead of edges, and take the subgraph to be the one induced by the chosen vertices.

Some terminology:

  • The edges are bonds
  • The vertices are sites
  • The chosen edges/vertices are open
  • The chosen graph is the open subgraph
  • The connected components of the open subgraph are open clusters

According to this language, the two models described above are bond percolation and site percolation.

The physical motivation for this model is a liquid percolating through a solid lattice, where the open bonds/sites represent the bonds/sites that the liquid can pass through. The main question suggested by this interpretation is whether the liquid will be able to pass through the lattice completely, which corresponds to the existence of an infinite open cluster.

The Harris-Kesten theorem mentioned above answers this question for bond percolation on {{\mathbb Z}^2}. It shows that for {p>\frac{1}{2}}, there is an infinite open cluster with probability 1, and for {p<\frac{1}{2}} there is an infinite open cluster with probability 0.

This post takes the first step toward proving this result by defining two critical probabilities that are closely related to this question, and then computing them for some simple graphs. Before we can even take that step though, we must construct a probability space {(\Sigma, \mathop{\mathbb P})} in which to work. For a graph {G=(V,E)}, we take:

  • {\Omega=\{0,1\}^{|E|}} to be the set of all arrangements of open edges, where 0 represents a closed edge and 1 represents an open edge; 
  • {\Sigma} to be the {\sigma}-algebra on {\Omega} generated by the cylinders

    \displaystyle C(F, \sigma)=\{\omega \in \Omega\mid \omega_f = \sigma_f\text{ for } f\in F\},\quad F\subset E\text{ and }\sigma =\{0,1\}^{|F|};

     

  • {\mathop{\mathbb P}} to be the probability measure on {\Sigma} induced by

    \displaystyle \mathop{\mathbb P}(C(F, \sigma))=\left(\prod_{\substack{f\in F\\ \sigma_f=1}} p\right)\left(\prod_{\substack{f\in F\\ \sigma_f=0}} 1-p\right).

In everything that follows, all probabilities are defined according to this measure.

2. Critical Probabilities

For a site {x}, let {C_x} be the open cluster containing {x}, and let {\theta_x(p)} be the probability that {C_x} is infinite.

Proposition 1. There is a critical probability {p_H} such that {\theta_x(p)=0} if {p<p_H} and {\theta_x(p)>0} if {p<p_H}, for all {x}.

(The notation {p_H} is in honor of Hammersley, who first proposed this model.)

Proof: Take any two sites {x, y} such that {|x-y|=d} (this is the distance induced by {G}). Then {\theta_x(p)\geq p^d \theta_y(p)}, since if there is an infinite open cluster containing {y} and a path from {y} to {x}, there will be an infinite open cluster containing {x} as well. Thus, if {\theta_x(p)=0} for one site, then {\theta_x(p)=0} for all sites, and if {\theta_x(p)>0} for one site, then {\theta_x(p)>0} for all sites. If we show that {\theta_x(p)} is an increasing function of {p}, the existence of {p_H} follows.

To see the increasing behavior (which is intuitively clear), consider assigning each edge {e} an independent random variable {X_e\sim \text{Uni}[0,1]}, and then taking it to be open if {X_e<p}. If {p_1<p_2}, the subgraph obtained with probability {p_1} will always be contained in the subgraph obtained with probability {p_2}, so {\theta_x(p)} must be increasing. \Box

We now connect this {p_H} to our original question about the existence of an infinite open cluster.

Corollary 2. Let {E} be the event that there is an infinite open cluster. Then {\mathop{\mathbb P}(E)=1} if {p>p_H} and {\mathop{\mathbb P}(E)=0} if {p<p_H}.

Proof: The event {E} is a tail event, so by the Kolmogorov 0-1 law, it can only take the values 0 and 1. If {p<p_H}, then {\mathop{\mathbb P}(E)\leq \sum_{\text{sites }x} \theta_x(p)=0,} so {\mathop{\mathbb P}(E)=0}. If {p>p_H}, then {\mathop{\mathbb P}(E)\geq \theta_x(p)>0} for at least one site, so {\mathop{\mathbb P}(E)=1}. \Box

So to understand when there is an infinite open cluster, it suffices to understand {p_H}.

When {p<p_H}, any {C_x} is almost surely finite, but it’s expected size might not be. Set {\chi_x(p)=\mathop{\mathbb E}(|C_x|)}. By the same arguments as before, {\chi_x(p)} increases with {p}, and it is either finite for all sites or for no sites. Thus we define

\displaystyle p_T=\sup\{p\mid \chi_x(p)<\infty\},

with the notation in honor of Temperley this time.

We have been deliberately ambiguous until now about whether we are working with bond percolation or site percolation in this section, and in fact we mean both. That is, there are actually four critical probabilities: {p_T^{\text{b}}, p_T^{\text{s}}, p_H^{\text{b}}, p_H^{\text{s}}}. And since each of these depends on the choice of graph, we actually need to write {p_T^{\text{s}}(G)}. This notation is a bit cumbersome though, so we refrain from using it except where absolutely necessary.

One obvious relationship between {p_T} and {p_H} is that {p_T\leq p_H}. Some more difficult relations are known, but we’ll look at those in a later post. For now, let’s do an example.

3. Computing the Critical Probabilities

Determining {p_T} and {p_H} in general is quite hard (after all, the Harris-Kesten theorem basically boils down to doing this for {{\mathbb Z}^2}, and is already highly non-trivial), but there are some simple graphs for which it can be done. We do an example with bond percolation.

Suppose {G=T_k} is the tree in which every vertex has degree {k+1}, except for the root which has degree {k}. Call the root {v_0} and let {T_{k,n}} be the subgraph obtained by truncating this graph {n} levels down from the root. We write {\pi_n} for the probability that, in bond percolation with probability {p}, there is an open path of length {n} from {v_0} to a leaf of {T_{k,n}}.

For there to be an open path of length {n} from {v_0} to a leaf of {T_{k,n}}, there must be an open edge between {v_0} and some other vertex, and then an open path from that vertex to a leaf. This gives

\displaystyle \pi_n=1-(1-p\pi_{n-1})^k.

Consider the function {f(x)=1-(1-px)^k}. The {\pi_n} result from iterating this function starting with {\pi_0=1}, so we need to study it’s fixed points. We have {f(0)=0} and {f(1)<1}. If {f'(0)>1}, then {f(\epsilon)>\epsilon} for some small {\epsilon>0}, but {f(1)<1}, so there is some fixed point {x_0}. Otherwise, the unique fixed point is at {x=0}. Concretely, {f'(0)=pk}, so this condition reduces to {p>\frac{1}{k}}.

In the case where {p>\frac{1}{k}}, we see that {\pi_{n-1}\geq x_0\implies \pi_n\geq x_0}, and since {\pi_0=1\geq x_0}, we always have {\pi_n\geq x_0}. But this {\pi_n} is a lower bound on {\theta_{v_0}(p)}, since if there is an infinite open cluster containing {v_0} there must be a path to the bottom, so {\theta_{v_0}(p)>0}, and so {p_H\leq \frac{1}{k}}.

On the other hand, if {p\leq \frac{1}{k}}, the {\pi_n} converge to the unique fixed point at 0, meaning that {\theta_{v_0}(p)=0}. It follows that {p_H=\frac{1}{k}} for {T_k}.

We can also do {p_T}: a site at depth {\ell} is part of {C_{v_0}} with probability {p^{\ell}}, so

\displaystyle \chi_{v_0}(p)=\sum_{y\in T_k} \mathop{\mathbb P}(y\in C_{v_0})=\sum_{\ell=0}^{\infty} k^{\ell}p^{\ell},

which is finite if and only if {pk<1}. Thus {p_T=\frac{1}{k}} as well.

A few remarks about this calculation: the fact that {p_T=p_H} here is not unusual. This is true for a very general class of graphs.

Also, it turns out that this tree {T_k} has, in some sense, the lowest critical probability possible. To see this, take {G} to be a graph with maximum degree {\Delta}. If {y\in C_x}, then there is an open path from {y} to {x}, so we can bound {\chi_x(p)} by the number of paths in {G} that start at {x}. Using our information about the maximum degree, this gives

\displaystyle \chi_x(p)\leq 1+\sum_{\ell=1}^{\infty} \Delta(\Delta-1)^{\ell-1} p^{\ell},

which converges exactly when {p<\frac{1}{\Delta-1}}. This means that {p_T\geq \frac{1}{\Delta-1}}, and since {p_H\geq p_T}, this bound holds for {p_H} as well. In {T_k}, {\Delta=k+1}, and this bound is attained.

So the tree {T_k} is special in more ways than one. It turns out that the more “tree-like” a graph is, the more easily {p_T} and {p_H} can be computed. Unfortunately, {{\mathbb Z}^2} is nothing like a tree, so a different approach is required to tackle it.

Q&A with the NYT Crossword Archive

I’ve been doing a lot of crosswords lately, so I thought I’d spend some time playing around with old crosswords to see if there was anything interesting to say about them. This whole post is structured as a series of questions I or other people asked, which I then answered using data from this repository. If there’s anything not covered here you’d like to know about, feel free to get in touch with me!

Just for reference, there are 1229405 clue/answer pairs, taken from 14524 puzzles. After throwing out duplicates, there are 140628 distinct answers, and 696050 distinct clues.

1. The Words

Question. What are the most common words? 
It turns out that these are almost exclusively short, vowel-rich words, with the top 5 being:

Word Occurrences
AREA 894
ERA 880
ERIE 818
ERE 750
ALOE 723

These aren’t especially interesting, so I then filtered by length i.e. found the most common word of each length. They’re a bit unwieldy to format here, but you can see them all here. The smaller ones are still boring, but the longer ones are cute, and the sudden uptick in frequencies for the final block is especially helpful: there just aren’t that many very long words appearing in crosswords, it seems.

Question. What is the longest word? 
This turns out to be a bit tricky to answer: doing the obvious sorting gives the answer of:

“TENNINEEIGHTSEVENSIXFIVEFOURTHREETWOONEITSTWOOOTWO,”

which is from the 12/30/2001 puzzle, which I couldn’t find a copy of. The second longest is the theme answer from this puzzle, which makes it clear that the previous one was probably a rebus too.

If you don’t count a rebus answer (which is reasonable), then there’s a whole slew of 23 letter words coming from a time when the Sunday was 23 by 23. If you don’t want to count those either, then you’re left with the pile of 21 letter words that span modern Sunday grids.

Question. What word has the most distinct clues?
The answer here turns out to be for the most part, the same as the common words:

Word Distinct Clues
ERA 462
ERIE 439
ONE 432
ARIA 424
ORE 405

Just to take a peek under the hood, all of the distinct clues for ERA can be seen here, and they seem to fall into three categories:

  • ERA like the time period
  • ERA like the baseball statistic
  • ERA like the Equal Rights Amendment

So, on a spiritual level, there’s only 3 clues for ERA. On the other hand, ONE actually has many meaningfully different clues, all of which are listed here. I wish there was some way to capture this sense of “spiritually different clues” and then find the answer with the most “spiritually diverse” clues, but I can’t think of a good algorithmic way to do it.

Question. What words have very few distinct clues?
I arbitrarily chose to look at words that are in the 10000 most common words but have fewer than 5 clues. The 5 most common (although there are only 9 in total) are:

Word Occurrences Clues
ORAD 60 4
AVISO 38 4
BRAC 32 2
HADON 31 4
OEN 29 4

The occurrences column basically shows that for a word to have fewer than 5 distinct clues, it really can’t appear all that often. If it does, it ends up having at least technically diverse clues, like those for ERA.

Question. How has the popularity of words changed over time?
I plotted this for 10 common words – there don’t seem to be any appreciable trends.

crosswords_over_time

Question. How does the distribution of letters in crossword answers compare to in normal English?
One graph shows it all:
diffs

Some of these make sense: A and E are over-represented because it’s easy to fill in grids with them. Others make no sense at all: why is H so under-represented?

Question. What are the hardest words?
Well, first you have to come up with some way to measure difficulty: I assigned values of 1, 2, 3, 4, 5, 6, 4 to the days of the week in order, keeping with the conventional wisdom that Sunday is about as hard as Thursday. Here are the 5 words from the 10,000 most common words that had the highest average value:

Word Occurrences
ONLEAVE 4.97
ALACARTE 4.93
OPALINE 4.85
ETAMINE 4.79
ONEONEONE 4.78

2. The Clues

Question. What are the most common clues?
No real pattern to these:

Clue Occurrences
Concerning 317
Jai ___ 256
Greek letter 250
Chemical suffix 225
Gaelic 216

It’s interesting to look at the answers these clues point to: “Concerning” has a handful, “Jai ___” has only one (ALAI), “Greek letter” and “Chemical suffix” both have a decent mix again, and “Gaelic” is only ever a clue for ERSE.

Question. What are clues that have many distinct answers? 
I’m not going to give a list of the top results here, because many of them are things like “See 17-Across” or “Theme of this puzzle,” which are not very interesting. Two results that are in the right vein though, are “Split” and “Nonsense”, which have 47 and 40 possible answers respectively, that are listed out here and here.

Question. What are clues that have only one answer? 
There are tons of these. A small sample:

Clue Occurrences
___-majesté 143
___ avis 132
Dies ___ 132
Sri ___ 102
__-Magnon 98

Note that I’m omitting “Jai ___” and “Gaelic” for variety’s sake. As suggested by this sample, nearly all of the one-answer clues are fill in the blanks, which makes sense.

3. The Days

Question. How does the length of words vary over the week?
Not much, as it turns out:

length_by_day.png

Question. How does the frequency of puns vary over the week?
A lot, as it turns out:

puns_days

Question. How does the density of vowels vary over the week? 
And we’re back to not much:

vowels_days.pngSo it seems like puns best capture the difficulty gradient of the week.

4. The Years

Question. How does the length of words vary over the years?
In strange and inconsistent ways:

year_length.png

One fact that may be relevant: Will Shortz began editing the puzzle in 1993. I’m not sure if this explains the jump, but it’s certainly a compelling story.

Question. How does the frequency of puns vary over the years?
People love them more every year:

puns_years.png

It’s interesting to note that there’s still a spike in the mid-90s, but this spike comes 2-3 years after Shortz became editor.

Question. How does the density of vowels vary over the years? 
Basically the opposite of the puns:

vowels_years.png Again, the huge drop occurs when Shortz takes over.

Memory in Speech Processing

This very short post is just to jot down an epiphany I had about a connection between two different approaches used in speech processing.

One popular kind of feature extracted from raw speech data is an MFCC. I don’t want to dwell too much on what exactly this is, but it basically involves taking two Fourier transforms (so it’s the spectrum of a spectrum), which works well, but is completely mystifying. The first Fourier transform makes sense, since sound waves are a periodic phenomenon, but the second one makes much less sense.

Another group of approaches that are growing in popularity are recurrent neural networks, which are basically networks that have some notion of “memory” built in to them. This means that inputs seen at previous times can be used for prediction for a present input, although details vary a lot depending on the particular implementation.

The punchline is that, on some level, these are doing similar things. Or rather, the MFCC is also relying on the idea that data from previous time points are influencing the current data point.

To see this, suppose our data are {x_1, x_2,\cdots,} and that they follow a model of the form

\displaystyle x_t=s_t+As_{t-D},

for some other stationary series {s_t} and constants {A, D}. The recurrent neural network would try to learn that {x_{t-D}} carries information about {x_t} (since it is designed to have access to {x_{t-D}}. What about the MFCC?

It’s helpful to directly compute the spectrum, starting with the auto-covariance. If {\gamma_s(h)=\text{cov}(s_{t+h}, s_t)} is the auto-covariance of {s_t}, then the auto-covariance of {x_t} is

\displaystyle \gamma_x(h)=\text{cov}(x_{t+h}, x_t)=\text{cov}(s_{t+h}+As_{t-D+h}, s_t+As_{t-D})

which, because the covariance is bilinear, becomes

\displaystyle (1+A^2)\gamma_s(h)+A\gamma_s(h+D)+A\gamma_s(h-D).

Then the theoretical spectrum is

\displaystyle f_x(\omega)=\sum_{h=-\infty}^{\infty} \gamma_x(h)e^{-2\pi i \omega},

which because of our earlier computation expands to

\displaystyle \sum_{h=-\infty}^{\infty}((1+A^2)\gamma_s(h)+A\gamma_s(h+D)+A\gamma_s(h-D)) e^{-2\pi i \omega h}.

This can be rewritten as

\displaystyle (1+A^2)f_s(\omega)+Ae^{2\pi i \omega D}\sum_{h=-\infty}^{\infty} \gamma_s(h+D)e^{-2\pi i \omega (h+D)}+Ae^{-2\pi i \omega D}\sum_{h=-\infty}^{\infty} \gamma_s(h-D)e^{-2\pi i \omega (h-D)},

and then simplified to

\displaystyle (1+A^2)f_s(\omega)+Af_s(\omega)(e^{2\pi i \omega D}+e^{-2\pi i \omega D})=(1+A^2+2A\cos (2\pi \omega D))f_s(\omega).

So the spectrum actually has a periodic component, and taking a second Fourier transform means sussing out this periodicity.

This means then, that MFCCs and recurrent neural networks are actually making similar assumptions about the data, or at least trying to exploit the same kind of structure. I like this similarity, especially because it suggests that thinking about the effect of previously seen data points is the “right” way to think about speech data.