# The Lightning Model

James T. Campbell111Corresponding Author<EMAIL_ADDRESS>Partially
supported by a University of Memphis Hardin Honors College Summer Research
Grant Alexandra Deane222Partially supported by an NSERC-USRA grant Anthony
Quas333Partially supported by an NSERC grant

###### Abstract

We introduce a non-standard model for percolation on the integer lattice
$\mathbb{Z}^{2}$. Randomly assign to each vertex $a\in\mathbb{Z}^{2}$ a
potential, denoted $\phi_{a}$, chosen independently and uniformly from the
interval $[0,1]$. For fixed $\epsilon\in[0,1]$, draw a directed edge from
vertex $a$ to a nearest-neighbor vertex $b$ if $\phi_{b}<\phi_{a}+\epsilon$,
yielding a directed subgraph of the infinite directed graph
$\overrightarrow{G}$ whose vertex set is $\mathbb{Z}^{2}$, with nearest-
neighbor edge set. We define notions of weak and strong percolation for our
model, and observe that when $\epsilon=0$ the model fails to percolate weakly,
while for $\epsilon=1$ it percolates strongly. We show that there is a
positive $\epsilon_{0}$ so that for $0\leq\epsilon\leq\epsilon_{0}$, the model
fails to percolate weakly, and that when $\epsilon>p_{\text{site}}$, the
critical probability for standard site percolation in $\mathbb{Z}^{2}$, the
model percolates strongly. We study the number of infinite strongly connected
clusters occurring in a typical configuration. We show that for these models
of percolation on directed graphs, there are some subtle issues that do not
arise for undirected percolation. Although our model does not have the finite
energy property, we are able to show that, as in the standard model, the
number of infinite strongly connected clusters is almost surely 0, 1 or
$\infty$.

2010 Mathematics Subject Classification: 60K35, 82B43

Keywords: percolation, integer lattice, phase transition, infinite clusters

##  1 Introduction

In this paper we introduce and establish some preliminary results about the
following family of non-standard models for percolation on the directed
integer lattice $\mathbb{Z}^{2}$. Randomly assign a potential to each vertex
in $\mathbb{Z}^{2}$, where the values are chosen independently and uniformly
from the interval $[0,1]$. Such an assignment is called a vertex
configuration; if $\phi$ is such a configuration and $a\in\mathbb{Z}^{2}$, we
designate the value of $\phi$ at $a$ by $\phi_{a}$. Fix $\epsilon\in[0,1]$,
and for nearest neighbor vertices $a$ and $b$, draw a directed edge from $a$
to $b$ if $\phi_{b}<\phi_{a}+\epsilon$, giving a subgraph of the nearest
neighbor graph on $\mathbb{Z}^{2}$. Thus, each vertex configuration gives rise
to an edge configuration, and there is a natural probability measure (the
push-forward of Lebesgue product measure on $[0,1]^{\mathbb{Z}^{2}}$) on the
edge configuration space. The Lightning Model refers to this family (for
$0\leq\epsilon\leq 1$), or perhaps a fixed member of this family, of
configuration spaces.

If there is an infinite path originating at the origin 0, we say the
configuration percolates weakly. Define the strong cluster of 0 to be the
strongly connected component containing 0, namely, the set of vertices $a$ for
which there is a directed path from 0 to $a$ and also a directed path from $a$
to 0. When this cluster is infinite, we say that the configuration percolates
strongly. It is immediate that strong percolation implies weak percolation.

We are interested mainly in the question: for which values of $\epsilon$ does
the Lightning Model percolate with positive probability?

In Section 3 we find an $\epsilon_{0}>0$ so that for
$0\leq\epsilon<\epsilon_{0}$, weak percolation fails to occur. While soft
arguments establish the existence of such an $\epsilon_{0}>0$, our arguments
(based on computing the spectral radius of certain operators) give an explicit
non-trivial lower bound. We also show that when $\epsilon>p_{\text{site}}$,
the critical probability for standard site percolation on $\mathbb{Z}^{2}$,
strong percolation occurs in the Lightning Model. The estimates in this
section are not sharp, and there is a substantial gap between $\epsilon_{0}$
and $p_{\text{site}}$. This leaves the question of determining precise
critical values $0<\epsilon_{w}\leq\epsilon_{s}<1$ so that for
$0<\epsilon_{w}$, weak percolation does not occur, and for
$\epsilon>\epsilon_{w}$, weak percolation occurs with positive probability.
Similarly for $\epsilon_{s}$ and strong percolation. We conjecture that
$\epsilon_{w}=\epsilon_{s}$.

In the standard (non-directed) site and bond percolation models, it is
straightforward to show using ergodicity and the finite energy condition that
for each $p$, there is $N_{p}\in\\{0,1,\infty\\}$ such that the number of
infinite clusters for a.e. configuration is $N_{p}$ (first established in
[NS81]). Furthermore a well-known argument of Burton and Keane ([BK89]) shows
that $N_{p}$ only takes the values 0 and 1 in these models. These questions
become far more subtle when studying strongly connected clusters in the
Lightning Model. Firstly the model lacks the finite energy condition, but more
seriously strong connectedness in directed graphs is a much more sensitive
property than connectedness in undirected graphs. For example, changing a
single edge can break a single strongly connected cluster into infinitely many
finite clusters. In Section 4, we establish that in the Lightning Model, the
number of strongly connected clusters is almost surely 0, 1 or $\infty$. Our
proof is strictly two-dimensional, and we do not know if the corresponding
statement holds in the higher-dimensional version of the model. In Section 5,
we present some open questions, and give further examples highlighting the
difficulties with Burton-Keane type arguments in directed settings. As
Grimmett ([Gri06]) indicates, “The Burton-Keane method is canonical of its
type, and is the first port of call in any situation where such a uniqueness
result is needed”. It seems that for this model, a different sort argument
must be used, which presents a truly interesting situation.

We use the term Lightning Model because the base idea (since modified to its
present form) originated as a simple model for lightning in which preferred
transitions are possible based on relative values of a potential. We
discovered later that in fact a model similar to the one in this paper, in
three dimensions, had been proposed by climatologists ([RPK+07]). Although
their paper focuses on basic geometric properties of simulations in finite
regions, the double connection confirmed our name choice.

##  2 Preliminaries

###  2.1 Basic Definitions, Paths, Clusters

Our base graph is the infinite directed nearest-neighbor graph
$\overrightarrow{G}$ whose vertex set is $\mathbb{Z}^{2}$, with edge set
$E=\\{(a,a\pm e_{i}):a\in\mathbb{Z}^{2},\,i=1,2\\}$, where $e_{1}$ and $e_{2}$
are the unit coordinate vectors. Adjacent vertices $a,b\in\mathbb{Z}^{2}$ are
called neighbors.

######  Definition 2.1.

The set $\mathbb{V}$ of _vertex configurations_ , is defined as

$\mathbb{V}=[0,1]^{\mathbb{Z}^{2}},$

where $[0,1]$ is the unit interval equipped with the usual topology. We equip
$\mathbb{V}$ with the product topology and the Borel $\sigma$-algebra
$\mathcal{B}$, and put Lebesgue product measure $\lambda$ on
$(\mathbb{V},\mathcal{B})$.

For a fixed vertex configuration $\phi\in\mathbb{V}$, we will write $\phi_{a}$
(or when clarity requires, $\phi(a)$) to denote the value of $\phi$ at vertex
$a$, and call it the _potential_ at $a$.

######  Definition 2.2.

The set of _edge configurations_ is defined as

$\mathbb{E}={\\{\,\raisebox{2.0pt}{ \leavevmode\hbox to2.98pt{\vbox
to2.98pt{\pgfpicture\makeatletter\hbox{\hskip 1.49165pt\lower-1.49165pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.29166pt}{0.0pt}\pgfsys@curveto{1.29166pt}{0.71336pt}{0.71336pt}{1.29166pt}{0.0pt}{1.29166pt}\pgfsys@curveto{-0.71336pt}{1.29166pt}{-1.29166pt}{0.71336pt}{-1.29166pt}{0.0pt}\pgfsys@curveto{-1.29166pt}{-0.71336pt}{-0.71336pt}{-1.29166pt}{0.0pt}{-1.29166pt}\pgfsys@curveto{0.71336pt}{-1.29166pt}{1.29166pt}{-0.71336pt}{1.29166pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\;,\,\to\,\\}}^{E},$

again equipped with the product topology and Borel $\sigma$-algebra
$\mathscr{B}$.

######  Definition 2.3.

For a parameter $\epsilon\in[0,1]$, define
$f_{\epsilon}:\mathbb{V}\to\mathbb{E}$ by

$f_{\epsilon}(\phi)=z,$

where for every edge $e=(a,b)$ in $E$,

$z_{e}:=z(a,b)=\begin{cases}\rightarrow&\text{ if
}\phi_{b}<\phi_{a}+\epsilon;\\\ \raisebox{2.0pt}{ \leavevmode\hbox
to2.98pt{\vbox to2.98pt{\pgfpicture\makeatletter\hbox{\hskip
1.49165pt\lower-1.49165pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.29166pt}{0.0pt}\pgfsys@curveto{1.29166pt}{0.71336pt}{0.71336pt}{1.29166pt}{0.0pt}{1.29166pt}\pgfsys@curveto{-0.71336pt}{1.29166pt}{-1.29166pt}{0.71336pt}{-1.29166pt}{0.0pt}\pgfsys@curveto{-1.29166pt}{-0.71336pt}{-0.71336pt}{-1.29166pt}{0.0pt}{-1.29166pt}\pgfsys@curveto{0.71336pt}{-1.29166pt}{1.29166pt}{-0.71336pt}{1.29166pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&\text{
otherwise.}\end{cases}$

We think of $\to$ as representing a directed edge and  representing the
absence of such an edge. Hence for a fixed $\phi$ and $\epsilon$,
$f_{\epsilon}(\phi)$ represents an infinite directed subgraph of
$\overrightarrow{G}$ with vertex set $\mathbb{Z}^{2}$, with a directed edge
from the vertex $a$ to the vertex $b$ iff the potential at $b$ is lower than
that at $a$, up to a tolerance of $\epsilon$. We often use standard
percolation terminology and call the directed edges in this subgraph open.

######  Definition 2.4.

For $\epsilon\in[0,1]$ define a probability measure $\mathbb{P}_{\epsilon}$ on
$(\mathbb{E},\mathscr{B})$ by

$\mathbb{P}_{\epsilon}(A):=\lambda(f_{\epsilon}^{-1}(A)),\quad
A\in\mathcal{B},$

where $\lambda$ is the Lebesgue product measure on $(\mathbb{V},\mathcal{B})$.

######  Remark.

An efficient way to describe our setup is to view the vertex potentials as a
family $\\{U_{a}\\}_{a\in\mathbb{Z}^{2}}$ of i.i.d. standard uniform random
variables. For fixed $\epsilon>0$, declare the directed nearest neighbor edge
$(a,b)$ to be $\epsilon$-open if $U_{b}<U_{a}+\epsilon$.

######  Definition 2.5.

By the Lightning Model we mean the aggregate of probability spaces
$(\mathbb{E},\mathscr{B},\mathbb{P}_{\epsilon})$, for $0\leq\epsilon\leq 1$.
We may also refer to the space with a fixed $\epsilon$ as the Lightning Model.

We want to study the (typical) component structure of edge configurations in
the Lightning Model, for which the following definitions are useful.

######  Definition 2.6.

A path from a to b in a configuration $f_{\epsilon}(\phi)$ is a sequence of
distinct vertices
$a=a_{0},a_{1},a_{2},\ldots,a_{n-1},a_{n}=b\in\mathbb{Z}^{2}$ such that
$a_{i}$ is a neighbor of $a_{i+1}$ and
$f_{\epsilon}(\phi)(a_{i},a_{i+1})=\,\to\,$ for all $0\leq i<n$. In this case
we write $a\overset{\epsilon}{\to}b$. If both $a\overset{\epsilon}{\to}b$ and
$b\overset{\epsilon}{\to}a$, we write $a\overset{\epsilon}{\leftrightarrow}b$,
and say that $a$ and $b$ are bi-directionally connected.

Note that in the case $a\overset{\epsilon}{\leftrightarrow}b$, there is no
requirement that the forward and backward paths are the reverse of each other.

######  Definition 2.7.

Let $a\in\mathbb{Z}^{2}$. We define the _strongly-connected component_ of $a$
in $f_{\epsilon}(\phi)$ to be

$\\{b\in\mathbb{Z}^{2}:a\overset{\epsilon}{\leftrightarrow}b\\}\;.$

This is also called the _strongly-connected cluster of a_.

For the remainder of this paper, whenever we discuss paths, clusters, etc.,
the vertex configuration $\phi$ is assumed to have been sampled from
$\mathbb{V}$ and the relations $\overset{\epsilon}{\to}$,
$\overset{\epsilon}{\leftrightarrow}$ have been generated by
$f_{\epsilon}(\phi)$ as above.

######  Definition 2.8.

Let $a\in\mathbb{Z}^{2}$. If
$\\{b\in\mathbb{Z}^{2}:a\overset{\epsilon}{\to}b\\}$ is infinite, then we say
that $a$ _percolates weakly_.

It is clear from the definition that $\\{\phi:a\text{ percolates weakly}\\}$
is measurable. König’s Lemma ([Kön27]) implies that $a$ percolates weakly if
and only if there is an infinite path starting from $a$.

######  Definition 2.9.

Let $a\in\mathbb{Z}^{2}$. If the strongly-connected component of $a$ is
infinite, we say that $a$ _percolates strongly_.

The set $\\{\phi:a\text{ percolates strongly}\\}$ is also measurable. We use
the phrase weak (resp., strong) percolation to mean weak (resp., strong)
percolation at the origin. We denote the events of weak percolation and strong
percolation by $\\{0\overset{\epsilon}{\to}\infty\\}$ and
$\\{0\overset{\epsilon}{\leftrightarrow}\infty\\}$, respectively.

###  2.2 Basic results

Percolation in the Lightning Model is monotone in $\epsilon$. Here is the
argument. We define the following partial order on $\mathbb{E}$:

######  Definition 2.10.

Given two edge configurations $v,w\in\mathbb{E}$, we say $v\preceq w$ if every
open edge of $v$ is also open in $w$.

It is clear that the maps $\\{f_{\epsilon}\\}$ are monotone in $\epsilon$ with
respect to this partial order; that is,

$\epsilon\leq\epsilon^{\prime}\;\implies f_{\epsilon}(\phi)\preceq
f_{\epsilon^{\prime}}(\phi)\;\forall\;\phi\in\mathbb{V}.$ (MP)

In other words if $\epsilon\,\leq\epsilon^{\prime}$ then the directed graph
defined by $f_{\epsilon}(\phi)$ is a subgraph of
$f_{\epsilon^{\prime}}(\phi)$. The following immediate corollary is stated as
a theorem because of its importance:

######  Theorem 2.11.

Monotonicity of Percolation Probability
Both $\mathbb{P}_{\epsilon}(\\{\text{weak percolation}\\})$ and
$\mathbb{P}_{\epsilon}(\\{\text{strong percolation}\\})$ are non-decreasing
functions of $\epsilon$.

###### Proof.

Let $W_{\epsilon}=\\{\phi\in\mathbb{V}\colon
0\overset{\epsilon}{\to}\infty\text{ in $f_{\epsilon}(\phi)$}\\}$ and
$S_{\epsilon}=\\{\phi\in\mathbb{V}\colon
0\overset{\epsilon}{\leftrightarrow}\infty\text{ in $f_{\epsilon}(\phi)$}\\}$.
It follows immediately from (MP) that for $\epsilon\leq\epsilon^{\prime}$,
$W_{\epsilon}\subseteq W_{\epsilon^{\prime}}$ and $S_{\epsilon}\subseteq
S_{\epsilon^{\prime}}$. ∎

We now show that 0 goes to infinity with the same probability that infinity
comes to 0\. This result hints that weak percolation could force strong
percolation.

######  Definition 2.12.

Denote $\\{v\in\mathbb{Z}^{2}:v\overset{\epsilon}{\to}0\\}$ as the attracting
set of the origin. If this set is infinite, we write
$\infty\overset{\epsilon}{\to}0$.

Observe that the event $\\{\infty\overset{\epsilon}{\to}0\\}$ is measurable.
Recall that weak percolation is the event
$\\{0\overset{\epsilon}{\to}\infty\\}$.

######  Theorem 2.13.

$\mathbb{P}_{\epsilon}(\\{0\overset{\epsilon}{\to}\infty\\})=\mathbb{P}_{\epsilon}(\\{\infty\overset{\epsilon}{\to}0\\}).$

###### Proof.

Given an edge $e=(a,b)$, its flip is the edge $\bar{e}=(b,a)$. Define the
mirror transformation $M:\mathbb{E}\to\mathbb{E}$ as sending each edge to its
flip:

$M(z)_{e}=z_{\bar{e}},\quad z\in\mathbb{E},\;e\in E.$

Let $I$ be the involution on vertex configurations defined by
$I:\mathbb{V}\to\mathbb{V}$, $I(\phi)_{a}=1-\phi_{a}$. It is easy to check
that $f_{\epsilon}(I(\phi))_{e}=M(f_{\epsilon}(\phi))_{e}$. Recalling that
$\lambda$ is Lebesgue product measure on $\mathbb{V}$, one also checks that
$\lambda\circ I^{-1}=\lambda$. It follows that

$\mathbb{P}_{\epsilon}\circ M^{-1}=\lambda\circ f_{\epsilon}^{-1}\circ
M^{-1}=\lambda\circ(M\circ f_{\epsilon})^{-1}=\lambda\circ(f_{\epsilon}\circ
I)^{-1}=\lambda\circ f_{\epsilon}^{-1}=\mathbb{P}_{\epsilon},$

so that $\mathbb{P}_{\epsilon}$ is invariant under $M$. To finish, observe
that the mirror image of the event $\\{0\overset{\epsilon}{\to}\infty\\}$ is
the event $\\{\infty\overset{\epsilon}{\to}0\\}$. ∎

##  3 Upper and Lower Bounds for Percolation

In this section we show that for sufficiently small $\epsilon$, weak
percolation almost surely fails to occur. We first give an elementary counting
argument which shows that weak percolation fails when $\epsilon=0$. Then we
sketch a soft argument showing that weak percolation also fails for some
positive $\epsilon$. This argument, however, does not give explicit non-
trivial lower bounds for such $\epsilon$, so we include a third argument,
based upon estimating the spectral radius for an appropriate linear
transformation, which gives a non-trivial lower bound. We note that it is
certainly not sharp.

Here is an argument showing that weak percolation fails when $\epsilon=0$. Fix
a path $0\to a_{1}\dots\to a_{n}$ in $\overrightarrow{G}$. This path is open
in $f_{0}(\phi)$ if and only if $\phi_{0}>\phi_{a_{1}}>\ldots>\phi_{a_{n}}$.
This last event has probability $1/(n+1)!$ since the values
$\phi_{0},\ldots,\phi_{a_{n}}$ are chosen independently and uniformly from
$[0,1]$. But since paths are simple, there are at most $4\cdot 3^{n-1}$ paths
of length $n$ starting from the origin. Hence the probability that at least
one of the paths is open is bounded above by $4\cdot 3^{n-1}/(n+1)!$, which
clearly tends to 0 as $n\to\infty$. Since weak percolation can occur only if
there is an open path of length $n$ (from the origin) for each $n$, we see
that weak percolation fails almost surely when $\epsilon=0$.

Here is a soft argument showing that weak percolation fails also for some
positive $\epsilon$. Fix a non-self-intersecting path with $nk$ edges (where
$k$ is to be fixed below and $n$ will be increased to $\infty$) and vertices
$a_{0}=0,\ldots,a_{nk}$. Then the probability that the path is open is bounded
above by the probability that each of the paths $a_{ik}\to\ldots\to
a_{ik+(k-1)}$ are open for $i=0,\ldots,n-1$. Since these events are
independent, the probability of this is
$\mathbb{P}_{\epsilon}(\\{a_{0}\to\ldots\to a_{k-1}\text{ is open}\\})^{n}$.
Next one sees that $\mathbb{P}_{\epsilon}(\\{a_{0}\to\ldots\to a_{k-1}\text{
is open}\\})$ is a continuous function of $\epsilon$, converging to $1/k!$ as
$\epsilon\to 0$. Now choose $k=7$ (so that $k!>3^{k}$) and fix an $\epsilon>0$
sufficiently small so that $\mathbb{P}_{\epsilon}(a_{0}\to\ldots\to
a_{6}\text{ is open})<3^{-7}$. Then the standard counting argument shows that
the probability there exists an open path of length $nk$ starting from the
origin approaches 0 as $n\to\infty$.

Next we give a more detailed argument which obtains an explicit lower bound by
computing the probability that a fixed path of length $n$ is open using an
iterated linear operator whose spectral radius may be accurately estimated.

######  Theorem 3.1.

Let $\epsilon_{0}=0.1481$. In the Lightning Model with
$\epsilon\leq\epsilon_{0}$, the probability of weak percolation is zero.

###### Proof.

Fix $\epsilon>0$, and consider a fixed path in $\overrightarrow{G}$ consisting
of vertices with potential values $x_{0},x_{1},x_{2},\ldots,x_{n}$. The path
will be open iff the values satisfy
$x_{0}\overset{\epsilon}{>}x_{1}\overset{\epsilon}{>}x_{2}\overset{\epsilon}{>}\cdots\overset{\epsilon}{>}x_{n}$,
where we write $x_{a}\overset{\epsilon}{>}x_{b}$ to indicate
$x_{b}<x_{a}+\epsilon$. Since the values at each of the vertices are chosen
independently and uniformly, the probability of the set of vertex
configurations satisfying
$x_{0}\overset{\epsilon}{>}x_{1}\overset{\epsilon}{>}x_{2}\overset{\epsilon}{>}\cdots\overset{\epsilon}{>}x_{n}$
is given by

$\int_{0}^{1}dx_{0}\int_{0}^{\min(x_{0}+\epsilon,1)}dx_{1}\int_{0}^{\min(x_{1}+\epsilon,1)}dx_{2}\ldots\int_{0}^{\min(x_{n-1}+\epsilon,1)}dx_{n}\;.$
(1)

We define now a related sequence of functions that lends itself to recursive
evaluation:

######  Definition 3.2.

Let $y\in[0,1]$. For $\epsilon>0$ and $n=0,1,\dots$, define

$F^{\epsilon}_{n}(y)=\int_{0}^{\min(y+\epsilon,1)}dx_{1}\int_{0}^{\min(x_{1}+\epsilon,1)}dx_{2}\ldots\int_{0}^{\min(x_{n-1}+\epsilon,1)}dx_{n}.$

$F^{\epsilon}_{n}(y)$ gives the probability of being able to continue $n$
steps from a vertex having value $y$, i.e., it is the probability that there
exists an open path of length $n$ beginning at a given vertex, conditioned on
that vertex having potential value $y$.

This last formula contains one fewer integral than appeared in expression (1),
and the probability that a fixed path of length $n$ is open is
$F^{\epsilon}_{n+1}(1)$.

######  Example 3.3.

It is enlightening to calculate the first few of these directly. Here we are
assuming that $\epsilon<1/2$.

$F^{\epsilon}_{0}(y)=1.$

$F^{\epsilon}_{1}(y)=\begin{cases}1&y\in(1-\epsilon,1];\\\ \epsilon+y&y\leq
1-\epsilon.\end{cases}$

$F^{\epsilon}_{2}(y)=\begin{cases}-\frac{\epsilon^{2}}{2}+\epsilon+\frac{1}{2}&y\in(1-\epsilon,1];\\\
-\frac{\epsilon^{2}}{2}+2\epsilon-\frac{1}{2}+y&y\in(1-2\epsilon,1-\epsilon];\\\
\frac{3\epsilon^{2}}{2}+2\epsilon y+\frac{y^{2}}{2}&y\leq
1-2\epsilon.\end{cases}$

These piecewise defined polynomials in $y$ are continuous, as the values agree
at the endpoints of each subinterval of definition.

The endpoints of the intervals of the piecewise definition are all of the form
$1-j\epsilon$, or 0, motivating the following definition.

######  Definition 3.4.

Let $M=\lceil\frac{1}{\epsilon}\rceil$. For $j=0,1,\ldots,M-2$, we define

$I_{j}=\left(1-(j+1)\epsilon,1-j\epsilon\right],$

and for $j=M-1$ define

$I_{M-1}=[0,1-(M-1)\epsilon].$

Then $\\{I_{0},I_{1},\dots,I_{M-1}\\}$ gives a partition of [0,1] into $M$
subintervals of length $\epsilon$ with perhaps the exception of $I_{M-1}$,
which has length at most $\epsilon$.

In Example 3.3, each of the calculated $F^{\epsilon}_{n}(y)$, restricted to
the interval $I_{j}$, is a polynomial in $y$ of degree $j$. We will now show
that this pattern holds for all $n$ and $\epsilon\in(0,1]$. We note that
$F^{\epsilon}_{n}(y)$ satisfies
$F^{\epsilon}_{n}(y)=\int_{0}^{\min(y+\epsilon,1)}F^{\epsilon}_{n-1}(x)dx$,
which motivates the following.

Define a linear transformation on (suitable) functions by

$\mathscr{L}^{\epsilon}f(x)=\int_{0}^{\min(x+\epsilon,1)}f(t)\,dt,$

and set

$\mathcal{F}^{\epsilon}=\big{\\{}f:[0,1]\to\mathbb{R}:\
f\raisebox{-2.15277pt}{$\big{|}$}_{I_{j}}\text{is a polynomial of degree }\leq
j\big{\\}},$

a finite dimensional vector space.

######  Lemma 3.5.

For any $\epsilon>0$, $\mathcal{F}^{\epsilon}$ is invariant under
$\mathscr{L}^{\epsilon}$.

###### Proof.

Let $s_{j}=1-j\epsilon$ so that $I_{j}=(s_{j+1},s_{j}]$ for $j<M-1$ and
$I_{M-1}=[0,s_{M-1}]$. If $\mathds{1}_{I_{j}}$ denotes the indicator function
of $I_{j}$, it is easily checked that if we define

$h_{j,i}(x)=\left(x-s_{j+1}\right)^{i}\cdot\mathds{1}_{I_{j}}(x),$

then the set of functions

$\\{h_{j,i}(x):j=0,1,\ldots,M-1\text{ and }i=0,1,\ldots,j\\}$

forms a basis for $\mathcal{F}^{\epsilon}$. Hence the lemma will follow if we
show that $\mathscr{L}^{\epsilon}h_{j,i}\in\mathcal{F}^{\epsilon}$ for each
$0\leq j<M$ and $0\leq i\leq j$.

We deal first with the case $j<M-1$. If $x\leq s_{j+2}$, then $x+\epsilon\leq
s_{j+1}$ so that $\big{[}0,\min(x+\epsilon,1)\big{]}$ does not intersect
$I_{j}$. It follows that $\mathscr{L}^{\epsilon}h_{j,i}$ is identically 0 on
$\bigcup_{k=j+2}^{M-1}I_{k}$.

If $x\geq s_{j+1}$, then $x+\epsilon\geq s_{j}$, so that

$\int_{0}^{\min(x+\epsilon,1)}h_{j,i}(t)\,dt=\int_{I_{j}}h_{j,i}(t)\,dt=\frac{\epsilon^{i+1}}{i+1},$

which is independent of $x$. That is, the restriction of
$\mathscr{L}^{\epsilon}h_{j,i}$ to $\bigcup_{k=0}^{j}I_{k}$ is a degree 0
polynomial.

If $x\in(s_{j+2},s_{j+1}]$, then it is straightforward to calculate

$\int_{0}^{\min(x+\epsilon,1)}h_{j,i}(t)\,dt=\int_{s_{j+1}}^{x+\epsilon}h_{j,i}(t)\,dt=\frac{1}{i+1}(x-s_{j+2})^{i+1}.$

That is, the restriction of $\mathscr{L}^{\epsilon}h_{j,i}$ to $I_{j+1}$ is
$\frac{1}{i+1}h_{j+1,i+1}$. Combining these, we have shown that

$\mathscr{L}^{\epsilon}h_{j,i}=\frac{1}{i+1}\left(\sum_{k=0}^{j}\epsilon^{i+1}h_{k,0}+h_{j+1,i+1}\right).$
(2)

Similarly if $j=M-1$ and $x\in[0,1]$, then $x+\epsilon>s_{M-1}$, and we have

$\mathscr{L}^{\epsilon}h_{M-1,i}(x)=\int_{I_{M-1}}h_{M,i}(t)\,dt=\int_{0}^{1-(M-1)\epsilon}\big{(}t-(1-M\epsilon)\big{)}^{i}\,dt,$

which shows that

$\mathscr{L}^{\epsilon}h_{M-1,i}=\frac{1}{i+1}\big{(}\epsilon^{i+1}-(M\epsilon-1)^{i+1}\big{)}=\frac{\epsilon^{i+1}-(M\epsilon-1)^{i+1}}{i+1}\sum_{k=0}^{M-1}h_{k,0}.$
(3)

∎

######  Remark.

Even though each $h_{j,i}$ is not continuous, one can easily check that after
applying $\mathscr{L}^{\epsilon}$ each resulting function is continuous. Also
notice that since $M=\lceil\frac{1}{\epsilon}\rceil$, $0\leq
M\epsilon-1<\epsilon$, so that all the coefficients of the matrix representing
$\mathscr{L}^{\epsilon}$ with respect to this basis are non-negative.

######  Corollary.

$F^{\epsilon}_{n}(y)\in\mathcal{F}^{\epsilon}$ for every $n\in\mathbb{N}$.

###### Proof.

$F^{\epsilon}_{0}(y)=1\in\mathcal{F}^{\epsilon}$ by definition. The result
follows by induction, since
$F^{\epsilon}_{n}(y)=\left((\mathscr{L}^{\epsilon})^{n}1\right)(y)$. ∎

In order to estimate the growth rate of $F^{\epsilon}_{n+1}(1)$, the
probability a fixed path of length $n$ is open, we may work entirely within
$\mathcal{F}^{\epsilon}$. Fix the ordered basis
$S=\\{h_{0,0},\,h_{1,0},\,h_{1,1},h_{2,0},h_{2,1},h_{2,2},\,\ldots,\,h_{M-1,M-1}\\}$
for $\mathcal{F}^{\epsilon}$ and let $A$ be the matrix representing
$\mathscr{L}^{\epsilon}$ with respect to $S$. $F^{\epsilon}_{0}$ is constant
function $1$ on the interval $[0,1]$, which we can write as the linear
combination $1=1h_{0,0}+1h_{1,0}+\ldots+1h_{M-1,0}$. Hence, its coordinate
vector with respect to our ordered basis is
$\begin{bmatrix}1&1&0&1&0&0&\cdots&0\end{bmatrix}^{T}$. Since
$F^{\epsilon}_{n}=(\mathscr{L}^{\epsilon})^{n}\,F^{\epsilon}_{0}$, the
coefficients of the function $F^{\epsilon}_{n}$ with respect to the basis are
given by

$A^{n}\begin{bmatrix}1&1&0&1&0&0&\cdots&0\end{bmatrix}^{T}.$

We want to evaluate the function this represents, at the value $y=1$.
Evaluating at $y=1$ means we are only interested in the values of the
resulting function on the interval $I_{0}$, which is given by the coefficient
of $h_{0,0}$, which we can get by simply taking the first entry of the
previous matrix product.

That is,

$F^{\epsilon}_{n}(1)=\begin{bmatrix}1&0&0&\cdots&0\end{bmatrix}\left(A^{n}\begin{bmatrix}1&1&0&1&0&0&\cdots&0\end{bmatrix}^{T}\right)\,.$
(4)

It follows that

$F^{\epsilon}_{n+1}(1)\leq C\|A^{n+1}\|\leq C\|A^{n}\|,$ (5)

for some constant $C$ (which may depend upon $\epsilon$).

Let us consider our current position. We wish to show that for some small
$\epsilon>0$, weak percolation does not occur (a.s.). The value
$F^{\epsilon}_{n+1}(1)$ is the probability that a fixed path of length $n$ is
open. If we let $\mu_{n}$ denote the number of paths of length $n$ (starting
at 0, say) in $\mathbb{Z}^{2}$, it is sufficient to show

$\lim_{n\to\infty}\;\mu_{n}\cdot F^{\epsilon}_{n+1}(1)\,=\,0\,.$ (6)

By sub-multiplicativity, $\mu_{n}^{1/n}$ is convergent; the limit is the
_connective constant_ , $\lambda$. It is known ([PT00]) that $\lambda\leq
2.679192495$. Recall that the spectral radius of $A$ is given by
$\rho(A)=\lim_{n\to\infty}\|A^{n}\|^{1/n}$, Moreover $\rho(A)$ is just the
maximum of the absolute values of the eigenvalues of $A$. Now if
$\lambda\cdot\rho(A)<1$ then (6) follows. Thus, to establish (6) it is
sufficient to find $\epsilon_{0}>0$ for which $\rho(A)<0.373246<1/\lambda$.
Finally, by monotonicity, weak percolation would fail a.s. for each
$0\leq\epsilon\leq\epsilon_{0}$.

Using equations (2) and (3), we may easily compute the matrix entries for $A$.
When $\epsilon_{0}=0.1481$, $A$ is a $28\times 28$ matrix whose spectral
radius is approximately $\rho(A)\approx 0.373079$ so that
$\lambda\cdot\rho(A)<1$ as required. ∎

######  Proposition 3.6.

When $\epsilon$ is greater than $p_{\text{site}}$, the critical probability
for the standard site percolation model in $\mathbb{Z}^{2}$, the Lightning
Model has positive probability of strong percolation.

###### Proof.

For $\phi\in\mathbb{V}$ and $\epsilon>0$ set
$S=S_{\phi,\epsilon}=\\{a\in\mathbb{Z}^{2}:\phi_{a}>1-\epsilon\\}$. By
$\epsilon$-tolerance, if $a$ and $b$ are neighbors in $S$, then both edges
$(a,b)$ and $(b,a)$ are present in $f_{\epsilon}(\phi)$. In particular, if $C$
is a (non-directed) cluster in $S$, then $C$ is contained in a single strongly
connected component of $f_{\epsilon}(\phi)$.

Fix $\epsilon>p_{\text{site}}$. Since the vertex potentials are independently
uniformly distributed, each $a\in\mathbb{Z}^{2}$ belongs to the random set $S$
with probability $\epsilon$ independently of all other vertices. Let $A$ be
the set of vertex configurations such that $S$ is infinite. By standard site
percolation, $\lambda(A)>0$. By the previous paragraph,
$f_{\epsilon}^{-1}\\{0\overset{\epsilon}{\leftrightarrow}\infty\\}\supset A$,
so that
$\mathbb{P}_{\epsilon}(\\{0\overset{\epsilon}{\leftrightarrow}\infty\\})>0$ as
required.

∎

Wierman ([Wie95]) established that $p_{\text{site}}<0.679492$, which therefore
gives an upper bound for the critical threshold for strong percolation in the
lightning model.

##  4 Number of Infinite Components

In this section, we show that for the Lightning Model, the number of infinite
(strong) clusters is almost surely 0, 1, or $\infty$. By the results in the
previous section, for sufficiently large $\epsilon$ the Lightning Model
strongly percolates, so in that case there is at least one cluster. Due to
ergodic considerations, we know that the number of clusters is almost surely
constant. However, we cannot at this time rule out the possibility that there
are infinitely many infinite strong clusters.

######  Definition 4.1.

For $\omega\in\mathbb{E}$, let $N_{\omega}$ denote the number of infinite
strong clusters in the configuration $\omega$.

We remark that $N_{\omega}$ is a measurable function of $\omega$. For a
natural number $n$, let $B_{n}$ denote
$\\{(x,y)\in\mathbb{Z}^{2}:\max\\{|x|,|y|\\}\leq n\\}$. For $r\in\mathbb{N}$
and $\omega\in\mathbb{E}$, which we think of a subgraph of
$\overrightarrow{G}$, the _restriction_ of $\omega$ to $B_{r}$, written
$\omega|_{B_{r}}$, denotes the induced subgraph of $\omega$ on the vertex set
$B_{r}$. For $\omega\in\mathbb{E}$ and $m<n<r$, let $C(m,n,r)(\omega)$ denote
the number of clusters in $\omega|_{B_{r}}$ that intersect both $B_{m}$ and
$B_{n}^{c}$.

######  Lemma 4.2.

For $\omega\in\mathbb{E}$, the limits

$\displaystyle\lim_{r\to\infty}C(m,n,r)(\omega),$
$\displaystyle\lim_{n\to\infty}\lim_{r\to\infty}C(m,n,r)(\omega)\text{ and}$
$\displaystyle\lim_{m\to\infty}\lim_{n\to\infty}\lim_{r\to\infty}C(m,n,r)(\omega)$

all exist (the first for all $n>m>0$ and the second for all $m>0$). Also
$N_{\omega}=\lim_{m\to\infty}\lim_{n\to\infty}\lim_{r\to\infty}C(m,n,r)(\omega)$,
so that $\omega\mapsto N_{\omega}$ is measurable.

###### Proof.

For now, let $n>m>0$ be fixed. For $r>n$, let $\leftrightarrow_{r}$ be the
equivalence relation on $B_{n}$ where $u\leftrightarrow_{r}v$ if there is a
directed path from $u$ to $v$ in $\omega|_{B_{r}}$ and a directed path from
$v$ to $u$ in $\omega|_{B_{r}}$. These equivalence relations are increasing.
That is, if $u\leftrightarrow_{r}v$, then $u\leftrightarrow_{r^{\prime}}v$ for
all $r^{\prime}>r$. Since there are finitely many equivalence relations on
$B_{n}$, they stabilize at some $r_{0}>n$. From that point onwards, the
sequence $(C(m,n,r)(\omega))_{r\geq r_{0}}$ does not change. (Prior to this
point, $C(m,n,r)(\omega)$ may increase as new connections added in the outer
layer can ensure that a cluster connects $B_{m}$ to $B_{n}^{c}$ (in a
bidirectional way); or decrease as new connections added in the outer layer
can merge two previously existing clusters). It follows that
$\lim_{r\to\infty}C(m,n,r)(\omega)$ exists. We see that
$\lim_{r\to\infty}C(m,n,r)(\omega)$ is the number of clusters in $\omega$ that
intersect both $B_{m}$ and $B_{n}^{c}$.

Given this, it is clear that $\lim_{r\to\infty}C(m,n,r)(\omega)$ is a non-
increasing function of $n$ (as if a cluster intersects $B_{n^{\prime}}^{c}$
for $n^{\prime}>n$, then it intersects $B_{n}^{c}$), so that
$\lim_{n\to\infty}\lim_{r\to\infty}C(m,n,r)(\omega)$ exists. This is the
number of infinite clusters in $\omega$ that intersect $B_{m}$ as a cluster is
infinite if and only if it intersects each $B_{n}^{c}$.

Finally we see $\lim_{n\to\infty}\lim_{r\to\infty}C(m,n,r)(\omega)$ is an
increasing function of $m$, so the limit as $m$ approaches $\infty$ also
exists, possibly taking the value $\infty$. From the above, we see that
$\lim_{m\to\infty}\lim_{n\to\infty}\lim_{r\to\infty}C(m,n,r)(\omega)$ is the
number of infinite clusters in $\omega$ as required.

Since $C(m,n,r)(\omega)$ is a measurable function of $\omega$ (as it depends
on finitely many coordinates), it follows that $N_{\omega}$ is measurable as
required. ∎

Here is the main theorem of this section:

######  Theorem 4.3.

For each $\epsilon>0$, there exists $k\in\\{0,1,\infty\\}$ such that for
$\mathbb{P}_{\epsilon}$-a.e. $\omega\in\mathbb{E}$, $N_{\omega}=k$.

For the proof of Theorem 4.3, we introduce a transformation on vertex
configurations that modifies the potential values only for vertices in a large
finite box centered the origin. The idea is that as a result of applying the
transformation, a large sub-box (again centered at the origin) will be forced
to be strongly connected in every configuration.

For $n\in\mathbb{N}$, recall that
$B_{2n}=\\{(x,y)\in\mathbb{Z}^{2}\colon\max(|x|,|y|)\leq 2n\\}$. Within such a
box, define the sequence of _layers_ $L_{0},L_{1},\ldots$, $L_{2n}$ by

$L_{i}=\\{(x,y)\in\mathbb{Z}^{2}:\max(|x|,|y|)=2n-i\\}.$

These are illustrated in Figure 1.

We now define our transformation. For $n\in\mathbb{N}$ and $\eta\in(0,1)$,
define $\Psi_{n}^{\eta}:\mathbb{V}\to\mathbb{V}$ by

$\Psi_{n}^{\eta}(\phi)(a)=\begin{cases}(1-\eta)^{i}\phi(a)&\ \ \ \text{if
$a\in L_{i}\,$;}\\\ \phi(a)&\ \ \ \text{otherwise,}\end{cases}$

for any vertex configuration $\phi\in\mathbb{V}$ and vertex
$a\in\mathbb{Z}^{2}$. The following lemma describes the useful properties of
$\Psi_{n}^{\eta}$.

######  Lemma 4.4.

Let $\epsilon>0$ be given. The transformations $\Psi_{n}^{\eta}$ defined above
have the following properties:

  1. 1.

For each $n\in\mathbb{N}$ and $\eta\in(0,1)$, if $A\subset\mathbb{V}$ has
positive measure, then $\Psi_{n}^{\eta}(A)$ also has positive measure.

  2. 2.

Suppose $n>\log(\tfrac{1}{\epsilon})$ and set
$\eta=\log(\tfrac{1}{\epsilon})/n$. For each $\phi\in\mathbb{V}$,
$f_{\epsilon}(\Psi_{n}^{\eta}(\phi))$ has the property that all of the edges
within the central sub-box, $B_{n}$, are bidirectionally connected. In
particular, the strongly connected component of the origin contains $B_{n}$.

  3. 3.

Suppose $n>\tfrac{1}{\epsilon}$ and let $\eta$ be as in 2. There exists a
universal constant $\delta>0$, independent of $\epsilon$ and $n$, so that the
probability that $\Psi_{n}^{\eta}$ doesn’t break any edges is at least
$\delta$. That is,

$\lambda\Big{(}\big{\\{}\phi\in\mathbb{V}\colon f_{\epsilon}(\phi)\preceq
f_{\epsilon}(\Psi_{n}^{\eta}(\phi))\big{\\}}\Big{)}\geq\delta.$

Figure 1: Layers $L_{0},L_{1},\ldots$ in the case $n=5$.

###### Proof of Lemma 4.4:.

Condition 1 follows since $\Psi_{n}^{\eta}$ is simply scaling the values of a
finite number of coordinates.

To establish condition 2, we show that $(1-\eta)^{n}<\epsilon$. This ensures
that for all $\phi\in\mathbb{V}$, the potential values of
$\Psi_{n}^{\eta}(\phi)$ at vertices in $B_{n}$ are all less than $\epsilon$.
By the argument in Proposition 3.6, this ensures that the edges contained in
$B_{n}$ are bidirectionally connected in
$f_{\epsilon}(\Psi_{n}^{\eta}(\phi))$.

Notice that $(1-\eta)^{n}<\epsilon$ follows immediately from the fact that for
$\eta\in(0,1)$, $1-\eta<e^{-\eta}$ and hence
$(1-\eta)^{n}<e^{-n\eta}=\epsilon$.

We move to the proof of 3. Define444This order induces the partial order
$\preceq$ on edge configuations in the obvious way. an order $\sqsubseteq$ on
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,\sqsubseteq\
\rightarrow\;$. Let $a,b\in\mathbb{Z}^{2}$ be any two adjacent vertices, and
let $\phi\in\mathbb{V}$ be some vertex configuration. We say that the edge
$e=(a,b)$ is _not broken_ by $\Psi_{n}^{\eta}$ if
$f_{\epsilon}(\phi)_{e}\sqsubseteq f_{\epsilon}(\Psi_{n}^{\eta}(\phi))_{e}$.

First notice that if $a,b\in\mathbb{Z}^{2}$ are adjacent vertices in the same
layer $L_{i}$, then the edge $(a,b)$ is never broken by $\Psi_{n}^{\eta}$.
This is because if both vertex values are scaled by the same positive value,
then the difference between them will be scaled by that value as well, and the
scaling is a contraction. It is worth noting that this may create new open
edges, something we will need to consider later in the proof of Theorem 4.3.

If $a$ and $b$ are adjacent vertices such that $a\in L_{i}$ and $b\in L_{i+1}$
(i.e. $b$ is closer to the origin), we refer to the edge $(a,b)$ as an
_inwards_ edge (similarly $(b,a)$ is an _outwards_ edge). We claim that if
$(a,b)$ is an inwards edge and $a\to b$ in $f_{\epsilon}(\phi)$, then $a\to b$
in $f_{\epsilon}(\Psi_{n}^{\eta}(\phi))$. Since $a\to b$ in
$f_{\epsilon}(\phi)$, we have $\phi(b)\leq\phi(a)+\epsilon$. This implies
$\Psi_{n}^{\eta}(\phi)(b)=(1-\eta)^{i+1}\phi(b)\leq(1-\eta)^{i+1}\phi(a)+(1-\eta)^{i+1}\epsilon\leq(1-\eta)^{i}\phi(a)+\epsilon=\Psi_{n}^{\eta}(\phi)(a)+\epsilon$,
so that $a\to b$ in $f_{\epsilon}(\Psi_{n}^{\eta}(\phi))$ also.

We now know that the only edges that might not be preserved by
$\Psi_{n}^{\eta}$ are the outwards edges. We will narrow this down even
further, proving that only edges in an outer “ring” near the boundary of
$B_{n}$ might not be preserved. This will be useful because it represents a
small quantity of the total number of edges in $B_{n}$.

######  Claim.

Let $e=(a,b)$ be an outwards edge with $a\in L_{i+1}$ and $b\in L_{i}$ and
$a\to b$ in some configuration $f_{\epsilon}(\phi)$. If
$i\geq\frac{1}{\epsilon}$ then $a\to b$ in
$f_{\epsilon}(\Psi_{n}^{\eta}(\phi))$.

That is, for $i\geq\frac{1}{\epsilon}$, no edges from level $i+1$ to level $i$
are broken.

###### Proof of Claim:.

By assumption $\phi_{b}<\phi_{a}+\epsilon$ and $i\geq 1/\epsilon$. We simply
need to verify that $(1-\eta)^{i}\phi_{b}<(1-\eta)^{i+1}\phi_{a}+\epsilon$.

We compute

$\displaystyle(1-\eta)^{i}\phi_{b}$
$\displaystyle<(1-\eta)^{i}\phi_{a}+(1-\eta)^{i}\epsilon$
$\displaystyle=(1-\eta)^{i+1}\phi_{a}+\epsilon+\eta(1-\eta)^{i}\phi_{a}-\epsilon(1-(1-\eta)^{i})$
$\displaystyle=(1-\eta)^{i+1}\phi_{a}+\epsilon+(1-\eta)^{i}\left[\eta\phi_{a}-\epsilon\left(\frac{1}{(1-\eta)^{i}}-1\right)\right].$

Since $\frac{1}{1-\eta}>1+\eta$, we see $\frac{1}{(1-\eta)^{i}}>1+i\eta$ so
that
$\eta\phi_{a}-\epsilon\big{(}\frac{1}{(1-\eta)^{i}}-1\big{)}<\eta\phi_{a}-\epsilon
i\eta=\eta(\phi_{a}-\epsilon i)<0$. Hence, the term displayed above in square
brackets is negative, so that

$(1-\eta)^{i}\phi_{b}<(1-\eta)^{i+1}\phi_{a}+\epsilon,$

as required. ∎

There is now a very small collection of edges which might not be preserved,
the outwards ($L_{i+1}\to L_{i}$) edges in the outermost $\frac{1}{\epsilon}$
layers ($i$ ranging between 0 and $\lfloor\frac{1}{\epsilon}\rfloor$). We move
to estimating the probability that they might be broken.

We call an edge between a vertex in $L_{1}$ and a vertex in $L_{0}$ an _outer_
edge. We begin by obtaining an upper bound on the probability that an outer
edge will not be preserved, and then use this to get an upper bound for the
probability that any edge is not preserved.

######  Claim.

Let $e=(a,b)$ be an outer edge. $\mathbb{P}_{\epsilon}(e\text{ is
broken})\leq\frac{\eta}{2}$.

###### Proof of Claim:.

Let $a\in L_{1}$ and $b\in L_{0}$ be neighbors in $\mathbb{Z}^{2}$. The
probability that the edge $(a,b)$ is not preserved by $\Psi_{n}^{\eta}$ is the
probability that the following two inequalities hold:

$\displaystyle\phi_{a}$ $\displaystyle>\phi_{b}-\epsilon;$
$\displaystyle\Psi_{n}^{\eta}(\phi)(a)$
$\displaystyle\not>\Psi_{n}^{\eta}(\phi)(b)-\epsilon.$

Let $s=\phi_{a}$ and $t=\phi_{b}$. The above conditions can then be written as

$\begin{split}t&<s+\epsilon;\\\ t&\geq(1-\eta)s+\epsilon.\end{split}$ (7)

The probability we want is the area of the region in $[0,1]^{2}$ defined by
these two inequalities. While the exact value is slightly messy to compute, we
may obtain a useful overestimate if we allow $t$ to extend to its maximum
value for $s\in[0,1]$. This region is a triangle with vertices at
$(0,\epsilon)$, $(1,1+\epsilon)$ and $(1,1-\eta+\epsilon)$, so it has width 1
and height $\eta$, and therefore area $\frac{\eta}{2}$. Hence, for an outer
edge $e$, $\mathbb{P}_{\epsilon}(e\text{ is broken})\leq\frac{\eta}{2}$, as
claimed. ∎

We now move to the case of a general outwards edge from $L_{i+1}\to L_{i}$,
for $1\leq i\leq\lfloor\frac{1}{\epsilon}\rfloor$. In order for an edge
$(a,b)$ with $a\in L_{i+1}$ and $b\in L_{i}$ to not be preserved, the
constraints (7) become

$\displaystyle t$ $\displaystyle<s+\epsilon;$ $\displaystyle t$
$\displaystyle\geq(1-\eta)s+(1-\eta)^{-i}\epsilon.$

Since the lower bound has been increased (as $\frac{1}{1-\eta}>1$), we see
that the area of the set of solutions is smaller than it was for the outer
edges from $L_{1}$ to $L_{0}$. Hence, for a general outward edge between
$L_{i+1}$ and $L_{i}$, the probability that it is broken is less than
$\eta/2$.

Summarizing: we know that the only edges which might be broken are outwards
edges in the outer $\lfloor\frac{1}{\epsilon}\rfloor$ layers, and that the
probability that any one of them is broken is less than $\eta/2$.

We move next to calculating the probability that no edge is broken by
$\Psi_{n}^{\eta}$. To do so, we consider separately the corners and sides of
the outermost $\lfloor\frac{1}{\epsilon}\rfloor$ layers of $B_{n}$. Let
$C_{1},\ldots,C_{4}$ denote the four corners of $B_{n}$ of size
$\lfloor\frac{1}{\epsilon}\rfloor\times\lfloor\frac{1}{\epsilon}\rfloor$ and
let $S_{1},\ldots,S_{4}$ denote the sides of $B_{n}$, that is, the regions of
size
$\lfloor\frac{1}{\epsilon}\rfloor\times(4n+1-2\lfloor\frac{1}{\epsilon}\rfloor)$
and
$(4n+1-2\lfloor\frac{1}{\epsilon}\rfloor)\times\lfloor\frac{1}{\epsilon}\rfloor$
between the corners. These regions contain the only edges that could be broken
by applying $\Psi_{n}^{\eta}$.

Because there are fewer than $\frac{1}{\epsilon^{2}}$ outward edges in each
corner, and the the probability that any given outward edge is broken is at
most $\frac{\eta}{2}$, the probability that some edge is broken is bounded
above by $\frac{\eta}{2\epsilon^{2}}$ and hence the probability that no edge
is broken in any given corner by $\Psi_{n}^{\eta}$ is at least
$1-\frac{\eta}{2\epsilon^{2}}$.

Next, we move to the sides. We intend to show that within each side, the
probability that no edge is broken by $\Psi_{n}^{\eta}$ is at least
$(1-\frac{\eta}{2\epsilon})^{4n}$. For this estimate we cannot simply use the
union bound as we did for the estimate in the corners; doing this naively
gives an upper bound for the probability that an edge is broken which is
greater than 1! The difficulty we face is the built-in dependence of the edges
in the lightning model: while the vertex potential values are independent, the
existence of an edge $a\to b$ affects the probability of an edge $b\to c$. We
want to use independence in some fashion, however. To obtain our desired
estimate, we split each side into disjoint pieces for which the event that
some edge is broken in one piece is independent of the event that some edge is
broken in another piece.

Decompose each side $S_{i}$ into $4n+1-2\lfloor\frac{1}{\epsilon}\rfloor$
disjoint slices, that is, outward paths of length $\lfloor 1/\epsilon\rfloor$
which have one vertex in each of the layers $L_{0},L_{1},\ldots L_{\lfloor
1/\epsilon\rfloor}$. The union of these slices contain all of the outward
edges in $S_{i}$. In each slice, the union bound implies the probability than
some edge is broken is at most $\frac{\eta}{2\epsilon}$, hence the probability
that no edge is broken in a given slice is at least
$1-\frac{\eta}{2\epsilon}$. For each slice, consider the event that no edge is
broken. These events are independent. Thus, we see the probability that no
edge is broken in $S_{i}$ is at least
$(1-\frac{\eta}{2\epsilon})^{4n+1-2\lfloor\frac{1}{\epsilon}\rfloor}$.

Since the $C_{i}$’s and the $S_{j}$’s are mutually disjoint, the above
argument shows that the probability that no edges are broken in any of the
corners or sides is at least

$\left(1-\frac{\eta}{2\epsilon^{2}}\right)^{4}\left(1-\frac{\eta}{2\epsilon}\right)^{16n}.$

Recalling that $\eta=(\log\frac{1}{\epsilon})/n$, the probability that no
edges are broken is at least

$\left(1-\frac{\log\frac{1}{\epsilon}}{2n\epsilon^{2}}\right)^{4}\left(1-\frac{\log\frac{1}{\epsilon}}{2n\epsilon}\right)^{16n}.$

Since $\epsilon$ is assumed to be fixed, then by using the well-known result
that $(1+\frac{a}{n})^{n}\to e^{a}$ we see that, for large $n$, the lower
bound for the probability that no edges are broken approaches

$e^{-8\log\frac{1}{\epsilon}/\epsilon}=\epsilon^{8/\epsilon},$

so that the probability that there no edges are broken is bounded below
uniformly in $n$. ∎

###### Proof of Theorem 4.3.

Fix $\epsilon>0$ so that strong percolation occurs. By ergodicity,
$N_{\omega}$ is constant, $\mathbb{P}_{\epsilon}$-almost surely. Suppose for a
contradiction that there are almost surely exactly $k$ infinite strongly-
connected components with $k\in\mathbb{N}\setminus\\{1\\}$.

Suppose $n>\log\tfrac{1}{\epsilon}$ and let $\delta>0$ be as given in Lemma
4.4 so that for all such $n$,

$\lambda\Big{(}\big{\\{}\phi\in\mathbb{V}\colon f_{\epsilon}(\phi)\preceq
f_{\epsilon}(\Psi_{n}^{\eta}(\phi))\big{\\}}\Big{)}\geq\delta.$

By continuity of measures, we may choose $N>0$ so that for any $n\geq N,$

$\lambda\left(\\{\phi\in\mathbb{V}\colon\text{$B_{n}$ intersects each infinite
component in $f_{\epsilon}(\phi)$}\\}\right)>1-\delta.$

Now fix $n\geq\max\\{\log\tfrac{1}{\epsilon},N\\}$ and consider the two events
whose probabilities are given in the previous two inequalities. Since the sum
of the probabilities is larger than 1, the intersection $E$ of these two
events has positive probability. Since the image under $\Psi_{n}^{\eta}$ of a
set of positive measure still has positive measure (Condition 1 of Lemma 4.4),
we have $\lambda(\Psi_{n}^{\eta}(E))>0$.

Suppose $\phi\in E$. Then $f_{\epsilon}(\phi)$ has $k$ infinite clusters, and
the box $B_{n}$ must contain vertices from each of them. Additionally, any
edge of $f_{\epsilon}(\phi)$ will still be present in
$f_{\epsilon}(\Psi_{n}^{\eta}(\phi))$ (possibly becoming bidirectional). It
follows that in $f_{\epsilon}(\Psi_{n}^{\eta}(\phi))$, the $k$ infinite
strongly-connected clusters originally present in $f_{\epsilon}(\phi)$ are
contained in a single infinite cluster, which we will denote by
$\mathcal{C}_{*}(\phi)$.

It appears, then, that what we have created is a set of vertex configurations
of positive measure ($\Psi_{n}^{\eta}(E)$) for which the resulting edge
configurations have a single infinite cluster. However, in directed
percolation it is possible to modify finitely many edges and create an
infinite strong cluster where there was none before; see example 5.1 below.

Hence, it is conceivable that $k-1$ additional infinite clusters were created
during the modification process, which would avoid the contradiciton that we
seek.

We can deal with this issue by defining a new map $\widehat{\Psi_{n}^{\eta}}$
by

$\widehat{\Psi_{n}^{\eta}}(\phi)_{a}=\begin{cases}\left(\Psi_{n}^{\eta}(\phi)\right)_{a}&\text{if
$a\in\mathcal{C}_{*}(\phi))$;}\\\ \phi_{a}&\text{otherwise.}\end{cases}$

Since there are only finitely many possibilities for
$\mathcal{C}_{*}(\phi)\cap B_{n}$, and all of them depend measurably on
$\phi$, there is a fixed set $\Lambda\subset B_{n}$ such that
$\mathcal{C}_{*}(\phi)\cap B_{n}=\Lambda$ for all $\phi$ in a positive measure
subset $A$ of $E$. A finite energy argument similar to that given in the proof
of Lemma 4.4(1) then shows that $\widehat{\Psi_{n}^{\eta}}(A)$ is of positive
measure.

Our final claim is that for any $\phi\in A$, $\widehat{\Psi_{n}^{\eta}}(\phi)$
has a unique infinite strongly connected cluster. To see this, let
$\mathcal{C}$ be any cluster for the edge configuration
$f_{\epsilon}(\widehat{\Psi_{n}^{\eta}}(\phi))$. If $\mathcal{C}$ intersects
$\mathcal{C}_{*}(\phi)$, then $\mathcal{C}\supset\mathcal{C}_{*}(\phi)$ (note
they may not be equal since they are generated from different potentials:
$\widehat{\Psi_{n}^{\eta}}(\phi)$ and $\Psi_{n}^{\eta}(\phi)$). On the other
hand, if $\mathcal{C}$ does not intersect $\mathcal{C}_{*}(\phi)$, then the
restriction of $f_{\epsilon}(\widehat{\Psi_{n}^{\eta}}(\phi))$ to
$\mathcal{C}$ is the same as the restriction of $f_{\epsilon}(\phi)$
restricted to $\mathcal{C}$, so that $\mathcal{C}$ is a finite cluster. We
have shown that for each $\phi\in A$, any cluster in
$f_{\epsilon}(\widehat{\Psi_{n}^{\eta}}(\phi))$ is either contained in
$\mathcal{C}_{*}(\phi)$ or is finite. Since $A$ has positive measure, this
contradicts our original assumption that there were exactly $k$ infinite
strong components almost surely, and we are done. ∎

We remark that the proof in this section is essentially two-dimensional. Lemma
4.4, part 2 makes essential use of the fact that $\eta=\Theta(1/n)$: this
guarantees that after applying $\Psi$, the central block is fully connected.
On the other hand, Lemma 4.4, part 3 requires that $\eta=O(1/n)$: this part
ensures that no edges are broken when the potential is transformed by
$\Psi_{n}^{\eta}$. The proof works by showing that the only edges potentially
broken are those within $1/\epsilon$ of the edge. In two dimensions, there are
$O(n)$ edges that are potentially broken and each has a probability of the
order of $\eta$ of breaking, allowing us to show that the probability that no
edges are broken is $\Theta(1)$.

##  5 Open Problems

It is natural to ask whether one can rule out the case of infinitely many
infinite strong clusters. The Burton-Keane theorem [BK89] is a well known
approach to this in the case of undirected graphs. The following examples show
that some things behave quite differently when dealing with directed graphs.

######  Example 5.1.

Creating an infinite strong cluster by modifying a single edge: Consider an
edge configuration with edges only on two parallel lines, corresponding to,
say, the lines $y=1$ and $y=2$ in $\mathbb{Z}^{2}$. Suppose the top line is a
source, that is, each vertex $(x,2)$ for $x\geq 0$ has a right-pointing edge
to $(x+1,2)$, and each vertex $(x,2)$ for $x\leq 0$ has a left-pointing edge
to $(x-1,2)$. Reverse the corresponding edges on the line $y=1$, and at every
fourth $x$-value, say, place a downward edge from $(x,2)$ to $(x,1)$. This has
no strongly connected infinite component, but adding a single edge from
$(0,1)$ to $(0,2)$ creates one.

######  Example 5.2.

Many splitting points give rise to the same boundary partition: Burton and
Keane’s proof works by considering _“splitting points”_ , that is places where
when a single vertex is removed from a configuration, an infinite cluster
splits into at least three separate infinite clusters. The proof counts
splitting points, showing that if they exist, their number grows
proportionally to the volume of a region by the ergodic theorem, while showing
they grow at most proportionally to the surface area of a region. For the
upper bound they study, in a large volume, which boundary points belong to
which infinite component when a splitting point is removed. Critically,
removing different splitting points gives rise to a different infinite
component structures. Unfortunately in the directed case, the removal of many
splitting points may give rise to the same infinite component structures.

This is illustrated schematically in Figure 2: when any of the (red) splitting
points is removed, the only infinite clusters are the bi-directional paths
connecting the circle to infinity.

Figure 2: A failure of Burton-Keane in directed graphs

Two more questions of interest:

  1. 1.

Does positive probability of weak percolation imply positive probability of
strong percolation? We conjecture that it does.

  2. 2.

The results prior to Section 5 carry over to higher dimensions. However, the
proof of Theorem 4.3 depends upon working in two dimensions. Does this result
hold for $d\geq 2$?

Data Availability Statement: Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

## References

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  * [RPK+07] J. A. Riousset, V. P. Pasko, P. R. Krehbiel, R. J. Thomas, and W. Rison, _Three-dimensional fractal modeling of intracloud lightning discharge in a New Mexico thunderstorm and comparison with lightning mapping observations_ , J. Geophys. Res. Atmospheres 112 (2007), [D15203].
  * [Wie95] J. C. Wierman, _Substitution method critical probability bounds for the square lattice site percolation model_ , Combin. Probab. Comput. 4 (1995), 181–188.

J.T. Campbell∗ (Corresponding author), Department of Mathematical Sciences,
University of Memphis, Memphis TN 38152

E-mail address, J.T. Campbell: jcampbllmemphis.edu

A. Deane, Department of Mathematics and Statistics, University of Victoria,
Victoria, BC V8W 2Y2 Canada

E-mail address, A. Deane: alexandradeaneuvic.ca

A. Quas, Department of Mathematics and Statistics, University of Victoria,
Victoria, BC V8W 2Y2 Canada

E-mail address, A. Quas: aquasuvic.ca