A density version for H\"aggstr\"om's theorem

TL;DR

This paper investigates whether, in invariant percolation on regular trees, there always exists an infinite self-avoiding path with a higher density of open edges than the overall edge probability p.

Contribution

It introduces a density version of H"aggstr"om's theorem, exploring the existence of paths with greater local density of open edges in invariant percolation.

Findings

Identifies conditions under which such paths exist

Provides counterexamples or proofs for specific cases

Advances understanding of percolation path densities

Abstract

Given invariant percolation on a regular tree, where the probability of an edge to be open equals $p$, is it always possible to find an infinite self-avoiding path along which the density of open edges is bigger then $p$?

Equations

$$D(\overline{x})=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\sum _{k=1}^{n}S(x_{k-1},x_k)$$

$$D(S)=\underset{\overline{x}}{\sup }D(\overline{x}).$$

$$D_d(p)=\underset{S}{\inf }\mathrm{esssup}D(S),$$

$$a(d,k)=1-\sqrt[k]{1-\frac{2}{d}}.$$

$$1-\sqrt[k]{1-\frac{2}{d}}\le \frac{1}{k+1}$$

$${\left(1-\frac{1}{k+1}\right)}^k\le 1-\frac{2}{d}$$

$$\left(\genfrac{}{}{0ex}{}{n}{an}\right){\epsilon }^{an}\le (2{\epsilon }^a{)}^n.$$

$$D(B)\le f_d(\epsilon ):=\frac{\log (2(d-1))}{\log (1/\epsilon )}.$$

$$D_{\infty }(x)=\underset{d\to \infty }{\lim }D\left(\frac{2}{d}x\right)$$

Full text

Let $S$ be an invariant percolation on the edges of the $d$-regular tree, where the probability of an edge being open equals $p$. We think of $S$ as an invariant process with values in $\{0,1\}$ (1 corresponds to open edges). For $\overline{x}=(x_{0},x_{1},x_{2},\dots)$ an infinite self-avoiding path, let $D(\overline{x})$ be the density of the percolation along $\overline{x}$, that is,

[TABLE]

and let

[TABLE]

In general, this is a random variable that is $F_{\infty}$-measurable, where $F_{\infty}$ is the tail $\sigma$-algebra. We may look at the essential supremum of this random variable and define

[TABLE]

where the infimum is taken over all invariant percolation distributions on the $d$-regular tree. For background on invariant percolation, see e.g. [1].

Obviously, $D_{d}(p)$ is monotone in $p$ and $D_{d}(p)\geq p$.

More generally we may ask

In his seminal paper [2], Olle Häggström proved that any invariant percolation on the $d$-regular tree, with marginal at least $\frac{2}{d}$, has an infinite cluster. In particular, we get that $D_{d}(\frac{2}{d})=1$ and specifically $D_{3}(\frac{2}{3})=1$.

More generally, define

[TABLE]

Notice that for $d=3$ and $2\leq k\leq 5$ we have $a(d,k)<\frac{1}{k}$, so we get that $D_{3}(p)>p$ for any $p\in[a(3,k),\frac{1}{k})$, but if $k\geq 6$ then $a(3,k)>\frac{1}{k}$, so we obtain no new information.

However, for $d\geq 4$ we have $a(d,k)<\frac{1}{k}$ for all $k$, so we get some that $D_{d}(p)>p$ for any $p\in\cup_{k=1}^{\infty}[a(d,k),\frac{1}{k})$.

In fact,

Now

[TABLE]

is equivalent to

[TABLE]

and the left hand side is decreasing (as a function of $k$) so the maximum is obtained for $k=1$ and it is $\frac{1}{2}\leq 1-\frac{2}{d}$. ∎

[TABLE]

Since there are $d(d-1)^{n-1}$ paths of length $n$ we get that when $a>\frac{\log(2(d-1))}{\log(1/\varepsilon)}$ the probability of a path with $an$ 1’s decays exponentially. We conclude that

[TABLE]

We now claim that if $0\leq p<q\leq 1$ and $q-p\leq\varepsilon$ then $D_{d}(q)-D_{d}(p)\leq f_{d}(3\varepsilon)$. This implies uniform continuity since $f_{d}(\varepsilon)\to 0$ when $\varepsilon\to 0$.

To show the claim, let $S$ be an invariant percolation with marginal $p$ and $B$ bernoulli percolation with marginal $3\varepsilon$. Let $S^{\prime}$ be their maximum. Then $S^{\prime}$ has marginal $p+(1-p)3\varepsilon\geq q$ (since we may assume that $p\leq 2/3$, for $p>2/3$ we have $D_{d}(p)=D_{d}(q)=1$). Therefore, with positive probability, there is an infinite path $\overline{x}$ such that the density of $s^{\prime}$ along $\overline{x}$ is at least $D_{d}(q)$. But the contribution of $B$ to the density of $\overline{x}$ is at most $f_{d}(3\varepsilon)$, so the density of $S$ along $\overline{x}$ is at least $D_{d}(q)-f_{d}(3\varepsilon)$. ∎

In particular, $D_{d}(p)\to 1$ as $p\to\frac{2}{3}$ so for some $a<\frac{2}{3}$ we have $D_{3}(p)>p$ for all $p\in[a,1)$. However, we still don’t know that $D_{3}(p)>p$ for all $0<p<1$ and specifically that $D_{3}(1/2)>(1/2)$.

When $d\to\infty$ we have that if $p=\frac{1}{d}+\frac{1}{d^{2}}$ we have $1-(1-p)^{2}=2p-p^{2}>\frac{2}{d}$ so again we have $D(p)\geq\frac{1}{2}$. This works for any fixed $k$, so if we define the limit

[TABLE]

we know that $D_{\infty}(x)\geq\frac{1}{k}$ for any $x>\frac{1}{k}$.

Note that the same methods apply to site percolation on regular trees. However, in that case, as $d\to\infty$ the threshold in Häggström’s theorem tends to $\frac{1}{2}$ rather then 0. Indeed, the tree is a bipartite graph and the partition into two sides is invariant, hence we can define a percolation that choose one of the sides with equal probabilities and then put 1s on this side and 0s on the other. This gives a marginal of $\frac{1}{2}$ and also density of $\frac{1}{2}$ along any self-avoiding path. This percolation is ergodic, but have a nontrivial tail $\sigma$-algebra.

We may also consider more general processes, i.e. not $\{0,1\}$-valued.

An interesting side question is this:

Remark: Häggström’s theorem was extended to nonamenable Cayley graphs [1], all the discussion above adapts to this set up.

Acknowledgements: thanks to Tom Hutchcroft for a useful comment.