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Ticmate is a Swedish company, and our goal is to become the leading European company for sales of international tickets and tours to all kinds of events and places. We believe information about the best events and experiences around the world should be accessible no matter where you are booking from – we speak english, and you pay in GBP. We only work with certified ticket sellers, usually the very best in each destination. Here you will find all the information you need and great tickets for musicals, tours, football matches, the hottest attractions and experiences in the world's largest cities! Read more about us and what we do at About Ticmate.
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On the front page you can find a list of the most popular things to do and see in Berlin.
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text:
Matteo Polisi #10 Vancouver TSS FC Rovers · 2018
Regular Season 14 1056 2 5 13 3 0 0
2018 Totals 14 1056 2 5 13 3 0 0
07/10/2019 L 0-1 Eastside FC 1 90 0 0 1 0 0 0
07/07/2019 L 0-6 at Portland Timbers U23 1 90 0 0 1 0 0 0
07/05/2019 W 3-2 at Victoria Highlanders FC 1 90 1 0 1 0 0 0
06/30/2019 L 1-3 at Eastside FC 1 90 0 0 0 0 0 0
This website is powered by SportsEngine's Sports Relationship Management (SRM) software, but is owned by and subject to the USL League Two privacy policy. ©2020 SportsEngine, Inc.

redpajama_set_name: RedPajamaCommonCrawl


text:
L'Oberstaufen Cup 2011 è stato un torneo professionistico di tennis maschile giocato sulla terra rossa. È stata la 20ª edizione del torneo, che fa parte dell'ATP Challenger Tour nell'ambito dell'ATP Challenger Tour 2011. Si è giocato a Oberstaufen in Germania dal 4 al 10 luglio 2011.

Partecipanti

Teste di serie

 Ranking al 20 giugno 2011.

Altri partecipanti
Giocatori che hanno ricevuto una wild card:
  Dieter Kindlmann
  Kevin Krawietz
  Borut Puc
  Marcel Zimmermann

Giocatori che sono passati dalle qualificazioni:
  Kamil Čapkovič
  Tiago Fernandes
  Leonardo Kirche
  Gabriel Trujillo Soler
  André Ghem (lucky loser)

Campioni

Singolare

 Daniel Brands ha battuto in finale  Andreas Beck, 6–4, 7–6(7–3)

Doppio

 Martin Fischer /  Philipp Oswald hanno battuto in finale  Tomasz Bednarek /  Mateusz Kowalczyk, 7–6(7–1), 6–3

Collegamenti esterni

ATP Challenger Tour 2011

redpajama_set_name: RedPajamaWikipedia


When your hearing aid goes on the fritz, you can get expert service and repair from Better Hearing Aid Service.
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Cornwall woman Sandra Clinch has been jailed for fatally stabbing her husband Alan with a pair of scissors in a row about tidying their house.
A jury at Truro Crown Court convicted Clinch, 49, of manslaughter on the grounds of diminished responsibility.
In court, she presented herself as the loving mother and wife whose temper occasionally got the better of her.
But softly-spoken Clinch was a violent woman, prone to screaming at strangers and neighbours and with a reputation for taking matters into her own hands.
She once drove her car at a teenage friend of one of her daughters, stabbed a former husband in the leg and wounded another in the hand.
Her children told the court she had also used a shower head, a porcelain bowl or a glass picture as weapons.
One young neighbour referred to the 49-year-old as "the Incredible Hulk".
On the day Alan Clinch was killed, Clinch was enraged that her quiet, gentle and slightly-built fourth husband would not immediately comply with her orders to tidy up the house.
She grabbed the scissors and plunged them into Mr Clinch's chest.
Clinch, whose five children were from four different fathers, wept at regular intervals in the dock.
She wiped the tears from her eyes as she recalled the way her late husband would put his arms around her to soothe her whenever she showed the first signs of rage.
Yet her gaze remained fixed away from the witness box as each of her children gave evidence against a mother prone, they said, to hitting, punching and kicking.
Pregnant with her first child at 15 while in care, Clinch married her first husband, Andrew Fazekas, in 1981.
The night before their wedding, she tore down Christmas decorations and cut the telephone cable after engaging in a violent row with her fiance.
Before their divorce three years later, Mr Fazekas had already written in his diary about how her temper resulted in her stabbing him in the hand.
Neighbours in the tiny community of Darite also heard Clinch shout, scream and throw things at her husband, who they said had appeared solemn and withdrawn in the six months prior to his death.
Yet further evidence, this time from Clinch's mother, offered a more disturbing insight into the mind of a troubled teenage girl.
The court heard how Clinch's mother highlighted her child's violent unpredictability, telling social services nearly four decades ago how she suspected her daughter would one day be responsible for murder.
Ken Wilkins of Esteem, which supports male victims of domestic violence in Cornwall, said: "There is still a perception in society today that regards domestic abuse against men as a bit of a joke and that can be very detrimental to men having the confidence to come forward."


Near Rimelan, Rojava/Northern Syria. October 2014 | Funeral ceremony for Tofan Dêrik, a 19-year-old YPG fighter. He died on October 15, 2014 during battles against the »Islamic State«.
Near Rimelan, Rojava/Northern Syria. October 2014 | Funeral ceremony for the YPG fighter Tofan Dêrik.
Qamishli, Rojava/Northern Syria. October 2014 | Family members mourn the death of Helepçe at »martyrs' cemetery«. On 11 March 2014 the 26-year-old Helepçe was killed by a suicide bomb attack in Qamishli.
Qamishli, Rojava/Northern Syria. October 2014 | Members of the »People's Protection Units« (YPG) stand guard at the »martyrs' cemetery«.
Qamishli, Rojava/Northern Syria. October 2014 | Relatives mourn at the grave of the 12-year-old Besir Remezan. He was shot dead during clashes with turkish military in the boarder region Qamishli/Nusaybin on Oktober 9, 2014.
Qamishli, Rojava/Northern Syria. October 2014 | The 13-year-old Sandi sits at the grave of her brother Besir Remezan.
Qamishli, Rojava/Northern Syria. October 2014 | Employees of a sewing cooperative.
Qamishli, Rojava/Northern Syria. October 2014 | A seamstress in the sewing cooperative.
Tal Maruf, Rojava/Northern Syria. December 2014 | Participants of the obligatory basic military training.
Tal Maruf, Rojava/Northern Syria. December 2014 | Participants of the obligatory basic military training during their exercises in the village Tal Maruf. The kurdish village was taken by ISIS for one day in the end of February 2014. The residents were forced to leave the village, shops and houses were burned and plundered and the holy buildings were blown up and destroyed.
Shermukh, Rojava/Northern Syria. December 2014 | Fighters of the People's Protection Units (YPG) taking a break near the frontline.
Shermukh, Rojava/Northern Syria. December 2014 | A YPG fighter observes ISIS positions at the front line.
Shermukh, Rojava/Northern Syria. December 2014 | A unit of the YPG near the front line.
Shermukh, Rojava/Northern Syria. December 2014 | YPG fighters discuss the current situation on the frontline.


WARSAW, Ind. – For the first time in three weeks, the Franklin College men's tennis team was in action, taking on Manchester University at the Warsaw Tennis and Fitness Club.
The Grizzlies (3-9, 0-2 HCAC) came up just short in their search for their first conference win of the year, falling to the Spartans (2-6, 2-2 HCAC) 6-3.
The Grizzlies top two players, Alex Hammer and Blake Amonett, were responsible for all three of the team's points on Saturday. The two teamed up at the top doubles spot for a 9-7 win over Isaac Collis and Spencer Thomas.
In singles play, Hammer handed Collis a 6-1, 6-1 defeat in a showdown of the number one singles players. Amonett picked up his second point of the day shortly after with a 6-4, 6-2 win over Connor Eichenauer at No. 2 doubles.
But the Grizzlies would come up short across the rest of the lineup, suffering one singles defeat in three sets and having two other singles matches decided by tie-breaker.
Franklin will look to get in the win column next Saturday (Apr. 21), when they welcome Earlham and Defiance to the Mickel-Hoover Tennis Complex for a pair of conference meetings. The Grizzlies will take on Earlham at noon, followed by a 5:00 p.m. showdown with Defiance.


text:
\section{Introduction and main results}
\label{sec:introduction}

We consider random fields on the lattice~$\Z^d$ in dimensions $d\geq2$, with a
finite state space $S=\{0,1,\dots,q-1\}$ per site for some $q\geq2$. Thus, the
configuration space for the models studied here is $\Omega = S^{\Z^d}$. This
space is endowed with a probability measure~$\Prob$, which we assume to be
translation-invariant. The results in this paper hold under different further
conditions on the measure~$\Prob$, which we will define and discuss first. To
do so, for any configuration $\omega \in \Omega$ and any $V \subset \Z^d$, we
will write $\omega_V$ for the configuration restricted to the set~$V$, that
is, $\omega_V$ is considered to be an element of $S^V$.

\begin{definition}[Finite-energy property]
	We say that the measure~$\Prob$ has the \emph{finite-energy property} if
	there exists an $h\in(0,1)$ such that for all states $s\in S$,
	\begin{equation}
	\label{finiteE}
	\begin{split}
		h
		&\leq \inf_{\sigma \in \Omega} \Prob\bigl( \omega(x)=s \bigm|
		\omega_{\Z^d \setminus \{x\}} = \sigma_{\Z^d \setminus \{x\}} \bigr)
		\\
		&\leq \sup_{\sigma \in \Omega} \Prob\bigl( \omega(x)=s \bigm|
		\omega_{\Z^d \setminus \{x\}} = \sigma_{\Z^d \setminus \{x\}} \bigr)
		\leq 1-h.
	\end{split}
	\end{equation}
\end{definition}

\begin{definition}[Markov property]
	We say that the measure~$\Prob$ has the \emph{Markov property} if the
	state of a site $x\in\Z^d$ depends only on the states of its nearest
	neighbours in~$\Z^d$ and not on the rest of the field, that is, if for all
	$x\in\Z^d$ and $s\in S$,
	\begin{equation}
	\label{Markov}
		\Prob\bigl( \omega(x) = s \bigm| \omega_{\Z^d \setminus \{x\}} \bigr)
		= \Prob\bigl( \omega(x) = s \bigm| \omega_{N_x} \bigr),
	\end{equation}
	where $N_x = \{ y\in\Z^d : |y-x| = 1 \}$ is the set of neighbours of~$x$.
\end{definition}

Observe that if the random field is Markovian, then by translation invariance
and~\eqref{Markov} it has the finite-energy property~\eqref{finiteE} if and
only if for a given vertex~$x$, each state~$s$ has strictly positive
probability regardless of the states of the neighbours of~$x$. The Markov
property implies the following ``boundary-$s$ Markov property'' for every
state $s\in S$, which will actually be sufficient for our purposes:

\begin{definition}[Boundary-$s$ Markov property]
	For a finite $V\subset\Z^d$, write $N_V = \bigcup_{x\in V} (N_x\setminus
	V)$ for the set of neighbours of~$V$. We say that~$\Prob$ has the
	\emph{boundary-$s$ Markov property} if for every finite $V\subset\Z^d$,
	given that all sites of~$N_V$ are in the state $s\in S$, the configuration
	on~$V$ is conditionally independent of the configuration on $\Z^d\setminus
	(V\cup N_V)$.
\end{definition}

A special case of the boundary-$s$ Markov property is the so-called
\emph{empty-boundary Markov property} as defined in~\cite{Grim06}, which is
the boundary-$0$ Markov property in our terminology.

Note that in the discussion so far, we have considered Markovian properties
under nearest-neighbour dependencies only. Indeed, in the conditional
probabilities~\eqref{Markov} we only consider nearest neighbours of~$x$, and
in our formulation of the boundary-$s$ Markov property, it suffices that the
nearest neighbours of~$V$ are in the state~$s$ to have independence
between~$V$ and the outside world. It is not difficult, however, to extend the
methods in this paper to random fields with more general dependencies between
sites, as long as these dependencies have finite range.

As a typical example of the kind of random fields we want to study, we
consider site percolation. In this case, $q=2$ and for given $0<p<1$, we take
the measure~$\Prob$ to be the Bernoulli product measure $\Prob_p$ such that
$\Prob_p(\omega(x) = 1) = p$ and $\Prob_p(\omega(x) = 0) = 1-p$ for each
$x\in\Z^d$. Since the states of different sites are independent in this case,
it is clear that~\eqref{finiteE} holds with $h = \min(p, 1-p)$, and the field
is obviously Markovian as well. For a given percolation configuration $\omega
\in \Omega$, we say that a site $x\in\Z^d$ is \emph{occupied} if $\omega(x) =
1$ and \emph{vacant} if $\omega(x) = 0$. For convenience, the same terminology
will be used for the states $0$ and~$1$ of any random field. This does not
attach any special meaning to the states $0$ and~$1$, since one can always
study the random fields obtained by permuting the states of~$S$.

Using the terminology introduced above, we write $C(x) = C(x,\omega)$ for the
occupied cluster of the site $x \in \Z^d$. That is, $C(x)$ is the set of
occupied sites (i.e., sites in state~$1$) that can be connected to~$x$ by a
nearest-neighbour path passing only through occupied sites. By convention, we
set $C(x)= \varnothing$ if $\omega(x) \neq1$ and we write $C = C(0)$ for the
cluster of the origin. In general, if $X \subset \Z^d$, then the number of
sites in~$X$ is denoted by $|X|$. In particular, the number of sites in the
occupied cluster of the origin will be denoted by~$|C|$. The results in this
paper hold under the assumption that the distribution of $|C|$ has an
exponential or stretched exponential tail.

\begin{definition}[Exponential tail]
	We say that the cluster-size distribution has an \emph{exponential tail}
	if the limit
	\begin{equation}
	\label{limit}
		\mu = \lim_{n\to\infty} \big[ \Prob(|C|=n) \big]^{1/n} =
		\lim_{n\to\infty} \big[ \Prob(n \leq |C| < \infty) \big]^{1/n}
	\end{equation}
	exists for some $0<\mu\leq1$.
\end{definition}

\begin{definition}[Stretched exponential tail]
	We say that the cluster-size distribution has a \emph{stretched
	exponential tail with exponent $\beta\in(0,1)$} if the limit
	\begin{equation}
	\label{superlimit}
		\nu = \lim_{n\to\infty} \big[ \Prob(|C|=n) \big]^{1/n^\beta} =
		\lim_{n\to\infty} \big[ \Prob(n \leq |C| < \infty) \big]^{1/n^\beta}
	\end{equation}
	exists for some $0<\nu<1$.
\end{definition}

We note that, for instance, for subcritical percolation the cluster of the
origin has an exponential tail with~$\mu$ strictly between $0$ and~$1$
\cite[Theorems 6.78, 8.61, 8.65]{Grim99}. The result~\eqref{limit} also holds
for critical and supercritical percolation, but with $\mu=1$, which indicates
that the distribution decays slower than exponentially. For supercritical
percolation, it is in fact known that $\Prob(|C|=n)$ decays like a stretched
exponential with $\beta=(d-1)/d$. This result appears in~\cite{ACC90,Cerf99}
for $d=2$, in~\cite{Cerf00} for $d=3$, and in~\cite{Cerf06} for $d\geq4$.

It is believed that the behaviour of the cluster-size distribution described
above is rather typical. That is, \eqref{limit} is expected to be true quite
generally for random fields in the non-percolating regime, i.e., when all
clusters are finite. This result follows for instance if one can show that
there exists a constant $A>0$ such that
\begin{equation}
\label{supermulti}
	\frac{\Prob(|C|=n+m)}{n+m} \geq A \, \frac{\Prob(|C|=n)}n \,
	\frac{\Prob(|C|=m)}m \quad \text{for all }n,m\geq1.
\end{equation}
See~\cite[p.~91]{KS78} for an example of this supermultiplicativity result in
the case of percolation, which is reproduced in~\cite[Lemma~6.102]{Grim99}.
Furthermore, it is believed that~\eqref{superlimit} holds quite generally in
the percolating regime with an exponent $\beta = (d-1)/d$.

An example of a binary random field which does not satisfy the Markov property
but does satisfy the boundary-$s$ Markov property both for $s=0$ and for
$s=1$, is the random-cluster model~\cite{Grim06}. Although this model is
defined in terms of edge occupation statuses rather than site occupation
statuses (as considered here), it is not difficult to adapt our methods for
the random-cluster model. Note that for the random-cluster model, the
exponential decay in~\eqref{limit} can be shown by adapting the proof of
Lemma~6.12 in~\cite{Grim99}, see the corrected version of Theorem~5.47
in~\cite{Grim06}. Theorem~5.47 in~\cite{Grim06} is stated in the more general
setting of finite-energy FKG measures satisfying the empty-boundary (i.e.,
boundary-$0$) Markov property. An important open problem for the
random-cluster model is whether $\mu<1$ in the subcritical regime (see e.g.\
\cite[Conjecture~5.54]{Grim06}).

\subsection{Pattern theorems}
\label{ssec:pattern}

Throughout this paper, we will use~$Q$ to denote the cube of diameter~$r$ at
the origin,
\begin{equation}
	Q = \{ x\in\Z^d : 0\leq x_i< r \text{ for all}~i = 1,2,\ldots,d \},
\end{equation}
where the diameter $r>0$ is considered to be fixed once and for all. The
\emph{extended cube}~$\overline{Q}$ is obtained by extending~$Q$ by~$1$ unit
in all directions, that is,
\begin{equation}
	\overline{Q} = \{ x\in\Z^d : -1\leq x_i< r+1 \text{ for all}~i =
	1,2,\ldots,d \}.
\end{equation}
The \emph{boundary} of the cube~$Q$ is defined as $\partial Q = \overline{Q}
\setminus Q$. Likewise, the boundary of~$\overline{Q}$ is defined as
$\partial\overline{Q} = \overline{ \overline{Q} } \setminus \overline{Q}$,
where, as before, $\overline{ \overline{Q} }$ is obtained by
extending~$\overline{Q}$ by~$1$ unit in all directions. While~$Q$ always
denotes the cube at the origin, we will write~$Q_x$ to denote the cube at the
site~$x$, that is, $Q_x = \{x+y : y \in Q\}$. The boundary of  this cube, the
extended cube at~$x$, and its boundary are defined analogously as $\partial
Q_x = \{x+y : y \in \partial Q\}$, $\overline{Q}_x = \{x+y : y \in
\overline{Q}\}$ and $\partial \overline{Q}_x = \{x+y : y \in \partial
\overline{Q}\}$.

\begin{definition}[Pattern]
	A \emph{pattern}~$P$ is a prescribed configuration of the states of the
	sites in the cube~$Q$, that is, $P = (P(x) : x\in Q)$ is an element of
	$S^Q$.
\end{definition}

\begin{figure}
	\centering
	\begin{picture}(191,125)
	 \put(0,0){\includegraphics{PatternOnC}}
	 \put(50,2){\makebox(0,0)[b]{\smash{site $x$}}}
	 \put(159,2){\makebox(0,0)[b]{\smash{origin}}}
	\end{picture}
	\caption{\label{fig:PatternOnC}A piece of a three-state ($q=3$)
	random-field configuration on~$\Z^2$, with sites in states $0$, $1$
	and~$2$ depicted in white, black and gray, respectively. The square
	highlights a pattern of diameter $r=3$ occurring on the occupied cluster
	of the origin at the site~$x$.}
\end{figure}

Suppose that~$P$ is a pattern. Then for a given configuration $\omega \in
\Omega$, we say that $P$ \emph{occurs at the site~$x$} if $\omega(x+y) = P(y)$
for all $y \in Q$. We say that~$P$ occurs at~$x$ \emph{on~$C$} if~$P$ occurs
at~$x$ and $\partial Q_x \subset C$. Thus, in our terminology, a pattern can
only occur on the cluster of the origin if it is completely surrounded by this
cluster. See Figure~\ref{fig:PatternOnC} for an illustration. The reason for
defining occurrences of patterns on~$C$ in this way, is to guarantee that a
pattern will always contribute the same number of sites to~$C$ whenever it
occurs on~$C$. This fact is crucial for the two-pattern theorems stated below.
However, the standard pattern theorem (Theorem~\ref{thm:pattern} below) still
holds if we weaken our definition of an occurrence on~$C$, for instance if we
say that a pattern~$P$ occurs at~$x$ on~$C$ if $P$ occurs at~$x$ and $\partial
Q_x \cap C \neq \varnothing$. This is obvious, because there must be more
occurrences of~$P$ on~$C$ under the weaker definition.

The standard pattern theorem states that for a given pattern~$P$, if the
cluster of the origin has size~$n$, then, for some $a>0$ sufficiently small,
it is very unlikely that~$P$ occurs less than~$an$ times on this cluster. This
statement is true whether or not one allows occurrences of patterns to
overlap. However, in this paper we will also study the ratio between the
numbers of occurrences of two distinct patterns, and for these results it is
important that patterns cannot overlap. We will avoid this by imposing a
condition on which occurrences of a pattern are counted. Let us now explain
this in more detail.

First of all, we note that there are patterns $P$ and~$P'$ for which overlaps
are ruled out by definition. If that is the case, we do not have to impose
restrictions on which occurrences of the patterns we count for the two-pattern
theorems (Theorems \ref{thm:twopattern}  and~\ref{thm:strongtwopattern}) to
hold. For generality, however, we will choose to consider only occurrences of
patterns at sites of the grid $V := (r+2)\Z^d + (1,1,\ldots,1)$. That is, we
denote by $N_P = N_P(\omega)$ the total number of occurrences of the
pattern~$P$ on~$C$ at distinct sites of~$V$, and we will study~$N_P$ rather
than the total number of distinct occurrences of~$P$ on~$C$. To explain our
choice of~$V$, we note that later on, we are going to replace occurrences
of~$P$ at sites of~$V$ by occurrences of~$P'$, and we want to guarantee that
this does not change the state of the origin from occupied to non-occupied.
This is why $V$ has been chosen such that the origin does not belong to~$Q_x$
for any $x \in V$.

Henceforth, we will write $\Prob_n$ for the measure $\Prob$ conditioned on the
event $\{|C|=n\}$, that is, $\Prob_n( \,\cdot\, ) := \Prob\bigl( \,\cdot\,
\bigm| |C|=n \bigr)$. In Section~\ref{sec:pattern} we will prove the following
result, which is slightly different from the pattern theorem for lattice
clusters appearing in~\cite{Madr99} because of the way we have defined~$N_P$:

\begin{theorem}[Pattern theorem]
\label{thm:pattern}
	Suppose that~$\Prob$ has the finite-energy property~\eqref{finiteE} and
	that the cluster-size distribution satisfies~\eqref{limit}. Let~$P$ be a
	pattern.  Then there exists an $a>0$ such that
	\[
		\limsup_{n\to\infty} \big[ \Prob_n(N_P \leq an) \big]^{1/n} < 1.
	\]
\end{theorem}

The main new results of this paper, however, concern ratios between the
numbers of occurrences of two distinct patterns $P$ and~$P'$ on~$C$. In
particular, we are interested in patterns~$P$ and~$P'$ such that one of these
patterns contributes one more site to the cluster of the origin than the other
pattern if it occurs on~$C$. We shall see that for such patterns, bounds on
the ratio between the number of occurrences of~$P$ and the number of
occurrences of~$P'$ lead directly to bounds on $\Prob(|C| = n+1) / \Prob(|C| =
n)$ if the cluster-size distribution is known to have an exponential tail.

To state our results, for a given pattern~$P$, we shall write $c_P$ for the
number of occupied sites in the pattern~$P$ which will be part of the cluster
of the origin whenever~$P$ occurs on~$C$ at some site. We also introduce the
notation
\begin{equation}
\label{PBox}
	\Prob\bigl( \Box P \bigr)
	:= \Prob\bigl( \omega(x) = P(x)\ \forall x\in Q, \omega(x) = 1\ \forall
	x\in \partial Q \bigr)
\end{equation}
for the ``probability of an occurrence of~$P$ at the origin surrounded by an
occupied cluster''.

Now suppose that the cluster-size distribution satisfies~\eqref{limit}. Then
we define, for a given pattern $P$,
\begin{equation}
	\gamma_P := \mu^{-c_P} \, \Prob\bigl( \Box P \bigr),
\end{equation}
and, for distinct patterns $P$ and~$P'$,
\begin{equation}
	\label{gamma}
	\gamma_{PP'} := \frac{\gamma_P}{\gamma_{P'}} =
		\frac{\mu^{c_{P'}} \, \Prob\bigl( \Box P \bigr)}
		{\mu^{c_P} \, \Prob\bigl( \Box P' \bigr)}.
\end{equation}
We believe that a law of large numbers holds for patterns, stating that there
exists a $\rho>0$ such that for any pattern~$P$, the number $N_P$ is
concentrated around $\rho \, \gamma_P$ for ``almost all'' configurations.
Although we cannot prove this, the following theorem (which we shall prove in
Section~\ref{sec:twopattern}) implies that if one can prove a law of large
numbers for one particular pattern, then it must hold for all patterns. It
states that for ``almost all'' configurations, the ratio $N_P / N_{P'}$ is
concentrated around $\gamma_{PP'} = \gamma_P / \gamma_{P'}$ for two distinct
patterns $P$ and~$P'$:

\begin{theorem}[Two-pattern theorem]
\label{thm:twopattern}
	Let $P$ and $P'$ be two distinct patterns. Suppose that~$\Prob$ satisfies
	the boundary-$1$ Markov property, and that the cluster-size distribution
	satisfies~\eqref{limit}. Then for all $\epsilon>0$ and $\gamma_{PP'}$ as
	defined in~\eqref{gamma},
	\[
		\limsup_{n \to \infty} \big[ \Prob_n( |N_P - \gamma_{PP'} \, N_{P'}|
		\geq \epsilon n ) \big]^{1/n} < 1.
	\]
\end{theorem}

A natural question is whether stronger bounds on the ratio $N_P/N_{P'}$ hold,
showing for instance that the difference between this ratio and $\gamma_{PP'}$
is at most of order $|C|^\alpha$ for some $0<\alpha<1$. Indeed, there are two
cases in which we have obtained such stronger bounds. The first case is the
case $c_P = c_{P'}$, that is, the case where the cluster size does not change
if one replaces an occurrence of $P$ on~$C$ by an occurrence of $P'$ on~$C$.
The second case is the case of stretched exponential decay of the cluster-size
distribution (i.e., the supercritical case for percolation). In that case, our
stronger bounds on $N_P - \gamma_{PP'} \, N_{P'}$ also lead to a stronger
version of the ratio limit theorem, see Corollary~\ref{cor:strongratio} below.
The stronger version of the two-pattern theorem reads as follows:

\begin{theorem}[Strengthened two-pattern theorem]
\label{thm:strongtwopattern}
	Consider a random field which has the boundary-$1$ Markov property, the
	finite-energy property~\eqref{finiteE} and a cluster-size distribution
	satisfying~\eqref{limit}. Suppose that $P$ and $P'$ are two distinct
	patterns, and let $\gamma_{PP'}$ be defined as in~\eqref{gamma}.  Then the
	following statements hold:
	\begin{enumerate}
	\item[(i)] If $c_P = c_{P'}$, then for all $\alpha > \ahalf$ and for every
	$\epsilon > 0$,
	\[
		\limsup_{n \to \infty} \big[ \Prob_n( |N_P - \gamma N_{P'}| \geq
		\epsilon n^\alpha) \big]^{1/n^{2\alpha-1}} < 1.
	\]
	\item[(ii)] If the cluster-size distribution has a stretched exponential
	tail with exponent~$\beta$, then for every $\alpha \geq \ahalf(1+\beta)$
	and every $\epsilon>0$,
	\[
		\limsup_{n \to \infty} \big[ \Prob_n( N_P \geq \gamma N_{P'} +
		\epsilon n^\alpha ) \big]^{1/n^{2\alpha-1}} < 1.
	\]
	\end{enumerate}
\end{theorem}

\paragraph{Remark.}
In Section~\ref{sec:twopattern} we show that
Theorem~\ref{thm:strongtwopattern}(ii) can be strengthened if the
supermultiplicativity result~\eqref{supermulti} holds.
Under~\eqref{supermulti}, if $c_{P'} < c_P$ and the cluster-size distribution
has a stretched exponential tail with exponent~$\beta$, one can show that for
$\alpha = (2-\beta)^{-1}$ there exists an $a_0>0$ such that for every $a\geq
a_0$,
\begin{equation}
	\limsup_{n \to \infty} \big[ \Prob_n(N_P \geq \gamma N_{P'} + a n^\alpha)
	\big]^{1/n^{2\alpha-1}} < 1.
\end{equation}

Examples where Theorems \ref{thm:pattern}, \ref{thm:twopattern}
and~\ref{thm:strongtwopattern}(i) apply are percolation, the Ising and Potts
models, and, in general, the random-cluster measure. For
Theorem~\ref{thm:strongtwopattern}(ii), stretched exponential decay is needed.

\subsection{Ratio limit theorems}
\label{ssec:ratio}

We can use our pattern theorems to prove that the ratio of the probabilities
that the occupied cluster of the origin has sizes $n+1$ and~$n$ converges
with~$n$. To prove this, one can in principle follow Kesten's argument
in~\cite{Kest63}, where a ratio limit theorem is derived from a pattern
theorem for self-avoiding walks. The same argument also appears in~\cite{MS93,
Madr99}. It requires that the probability of a configuration changes by a
constant factor whenever one replaces a single occurrence of a pattern~$P$ on
the cluster of the origin by an occurrence of another pattern~$P'$. In our
present context, this means that we need to assume the boundary-$1$ Markov
property. However, under this condition we can use our two-pattern theorem
(Theorem~\ref{thm:twopattern}) to give a more elegant and direct proof of the
ratio limit theorem, avoiding the rather technical argument by Kesten. Our
proof is presented in Section~\ref{sec:ratio}.

\begin{corollary}[Ratio limit theorem]
\label{cor:ratio}
	Suppose that~$\Prob$ has the finite-energy property~\eqref{finiteE} and
	the boundary-$1$ Markov property, and that~\eqref{limit} holds for the
	cluster-size distribution. Then the limit~$\mu$ in~\eqref{limit} also
	satisfies
	\[
		\lim_{n\to\infty} \frac{\Prob(|C| = n+1)}{\Prob(|C| = n)} =
		\lim_{n\to\infty} \frac{\Prob(n+1 \leq |C| < \infty)}{\Prob(n \leq |C|
		< \infty)} = \mu.
	\]
\end{corollary}

Examples where Corollary~\ref{cor:ratio} applies are percolation, Ising and
Potts models, as well as the random-cluster model for general $p$ and $q$.
Note that in the case of percolation, for subcritical~$p$, the result in
Corollary~\ref{cor:ratio} is stronger than~\eqref{limit}, but still much
weaker than the widely believed tail-behaviour of the cluster-size
distribution, namely that there exist $\theta = \theta(d) \in \R$ and $A =
A(p,d)$ such that
\begin{equation}
	\Prob_p(|C| \geq n) = A n^\theta \mu^n [1+o(1)].
\end{equation}
For supercritical~$p$, we can obtain a stronger result than
Corollary~\ref{cor:ratio} by virtue of our stronger version of the two-pattern
theorem, Theorem~\ref{thm:strongtwopattern}:

\begin{corollary}[Strengthened ratio limit theorem]
\label{cor:strongratio}
	Suppose that~$\Prob$ has the boundary-$1$ Markov property and the
	finite-energy property~\eqref{finiteE}, and that the cluster-size
	distribution has a stretched exponential tail with exponent $\beta \in
	(0,1)$. Then, for every $\epsilon>0$,
	\[
		\left| \frac{ \Prob(|C| = n+1) }{ \Prob(|C| = n)} - 1 \right|
		\leq \frac{\epsilon}{n^{(1-\beta)/2}}
	\]
	for sufficiently large~$n$. Hence, for every $x>0$ and $0 < \alpha \leq
	(1-\beta)/2$,
	\[
		\lim_{n\to\infty} \frac{\Prob(|C| = n + \lfloor xn^\alpha
		\rfloor)}{\Prob(|C| = n)} = 1.
	\]
\end{corollary}

\paragraph{Remark.}
If the supermultiplicativity result~\eqref{supermulti} holds, then we can
derive a stronger lower bound for the ratio $\Prob(|C|=n+1) / \Prob(|C|=n)$,
by virtue of the Remark below Theorem~\ref{thm:strongtwopattern}. Namely, in
that case we can show that there exists a constant $A>0$ such that for~$n$
sufficiently large,
\begin{equation}
	\frac{ \Prob(|C| = n+1) }{ \Prob(|C| = n)} \geq 1 -
	\frac{A}{n^{(1-\beta)/(2-\beta)}}.
\end{equation}

Examples where Corollary~\ref{cor:strongratio} applies are supercritical
percolation, the Ising model and random-cluster measures in those cases where
the Wulff-shape results have been proved (see \cite{ACC90, Cerf99, Cerf00,
Cerf06} and the references therein).

\subsection{Consequences for maximal clusters}
\label{ssec:Gumbel}

In this subsection, we describe the consequences of the ratio limit theorem
for maximal clusters as derived in~\cite{HR06}. In order to state our results,
we need some further notation. Let $B_n = [-n,n]^d \cap \Z^d$ be the cube of
width $2n+1$. We let
\begin{equation}
\label{meenzb}
	|\Cmax| = |\Cmax(\omega)| = \max_{x\in B_n}|C(x)|
\end{equation}
denote the size of the maximal cluster having a non-empty intersection
with~$B_n$. Furthermore, we define the cluster $\Cle(x)$ by
\begin{equation}
\label{Cle}
	\Cle(x) = \begin{cases}
		C(x) & \text{if $x$ is the left-endpoint of~$C(x)$}, \\
		\varnothing & \text{otherwise},
	\end{cases}
\end{equation}
where by the left-endpoint of a finite set $A\subset\Z^d$, we mean the minimum
of~$A$ in the lexicographic order. In~\cite{HR06}, results were shown for
$|\Cmax|$ assuming the ratio limit theorem for the cluster $\Cle(0)$ instead
of $C(0)$. Therefore, we shall need the following corollary, which we shall
prove simultaneously with Corollary~\ref{cor:ratio} in
Section~\ref{sec:ratio}:

\begin{corollary}[Ratio limit theorem for~$\Cle(0)$]
\label{cor:ratioCle}
	Suppose that~$\Prob$ has the finite-energy property~\eqref{finiteE} and
	the boundary-$1$ Markov property, and that~\eqref{limit} holds. Then the
	limit~$\mu$ in~\eqref{limit} also satisfies
	\[
		\lim_{n\to\infty} \frac{\Prob(|\Cle(0)| = n+1)}{\Prob(|\Cle(0)| = n)}
		= \lim_{n\to\infty} \frac{\Prob(n+1 \leq |\Cle(0)| < \infty)}{\Prob(n
		\leq |\Cle(0)| < \infty)} = \mu.
	\]
\end{corollary}

The results for~$|\Cmax|$ in~\cite{HR06} hold under further conditions on the
measure~$\Prob$, that we will formulate now. We start by introducing a
so-called `high mixing' condition. For $A\subseteq \Z^d$, we write $E_{A}$ for
an event depending only on the site variables in~$A$. Let $\mathcal{F}_{A}$
denote the $\sigma$-field generated by the site variables in~$A$. For $m>0$,
we define
\begin{equation}
	\label{mixfun}
	\phi(m) = \sup \frac1{|A_1|} \, \bigl| \Prob\left(E_{A_1} \mid E_{A_2}
	\right) - \Prob\left(E_{A_1}\right) \bigr|,
\end{equation}
where the supremum is taken over all finite subsets $A_1,A_2$ of~$\Z^d$, with
$d(A_1,A_2)\geq m$ ($d$ denoting Euclidian distance) and over all $E_{A_i} \in
\mathcal{F}_{A_i}$ with $\Prob(E_{A_2}) > 0$.

Note that this $\phi(m)$ differs from the usual $\varphi$-mixing function
since we divide by the size of the dependence set of the event~$E_{A_1}$. This
is natural in the context of Gibbsian random fields, where the classical
$\varphi$-mixing mostly fails (except for the simplest i.i.d.\ case and ad-hoc
examples of independent copies of one-dimensional Gibbs measures). We are now
ready to formulate the non-uniformly exponentially $\phi$-mixing (NUEM)
condition:

\begin{definition}[NUEM]
	We say that a random field is \emph{non-uniformly exponentially
	$\phi$-mixing} (NUEM) if there exist constants $C,c>0$ such that
	\begin{equation}
	\label{exponmix}
		\phi(m) \leq C \exp(-c m) \quad \text{for all } m>0.
	\end{equation}
\end{definition}

Examples of random fields satisfying the NUEM condition are Gibbs measures
with exponentially decaying potential in the Dobrushin uniqueness regime, or
local transformations of such measures. Of course, for site percolation, where
we have independence, we have $\phi \equiv 0$. The NUEM condition is weaker
than the more often used weak mixing (see \cite{Alex98,Alex04} for a
definition of weak mixing and the related stronger notion of ratio weak
mixing). Weak mixing holds for (i) the Ising model on $\Z^d$ for supercritical
temperatures (\cite[Corollary~3.8]{Alex98}); (ii) the Ising model on $\Z^2$
for all temperatures and non-zero external field
(\cite[Corollary~3.8]{Alex98}); (iii) the Potts model on $\Z^2$ with $q\geq
26$ (\cite[Corollary~3.9]{Alex98} and \cite[Theorem~1.8]{Alex04}); for general
Potts models on $\Z^d$ under the assumption of exponential decay of
connectivities and random-cluster uniqueness; (iv) general random-cluster
measures on $\Z^d$ under the assumption of exponential decay of connectivities
and random-cluster uniqueness (\cite[Theorem~1.6]{Alex04}). We refer to
\cite{Alex98, Alex04} and the references therein for a discussion on mixing
aspects.

In the subcritical case, we shall deal with models where the cluster-size
distribution has exponential tails, i.e., we shall assume that $\Prob$, the
law of the random field, satisfies~\eqref{limit} with $\mu<1$. We shall also
assume a \emph{second moment condition}, which is used in~\cite{HR06} to prove
the asymptotics of the largest connected component. The precise assumption is
that for all $\alpha>1$,
\begin{equation}
\label{secmcon}
	\limsup_{n\to\infty} \sum_{0<|x|<n^\alpha}
	\frac{\Prob\bigl( n\leq |\Cle(x)| < \infty,\,
			n\leq |\Cle(0)| < \infty \bigr)}
	{\Prob\bigl( n\leq |\Cle(0)| < \infty \bigr)} < \infty,
\end{equation}
In \cite[Proposition~3.7]{HR06}, it is shown that for Markov models with the
FKG property, \eqref{secmcon} follows from~\eqref{limit} with $\mu<1$. An
inspection of the proof of \cite[Proposition~3.7]{HR06} shows that it also
applies to models with the FKG property satisfying the boundary-$s$ Markov
property with $s=0$.

Let us now state our results for~$|\Cmax|$. As an implication of
Corollary~\ref{cor:ratioCle}, we obtain the following result on the
subcritical maximal cluster:

\begin{theorem}[Subcritical Gumbel maximal cluster]
\label{thm:Gumbel}
	Assume that $\Prob$ has the finite-energy property~\eqref{finiteE}, is
	NUEM, and satisfies \eqref{limit} with $\mu<1$ and~\eqref{secmcon}. Then
	there exists a sequence $u_n\in\N$, with $u_n\to\infty$, real numbers $a,
	\rho>0$ and a bounded sequence $a_n\in[a,1]$, such that for all $x\in\N$,
	\[
		\Prob(|\Cmax| \leq u_n + x) = e^{-a_n \, \mu^x} + O(n^{-\rho}).
	\]
\end{theorem}

Theorem~\ref{thm:Gumbel} shows that $|\Cmax|$ is bounded above and below by
Gumbel laws, and shows in particular that the sequence $|\Cmax| - u_n$ is
tight. As explained in more detail in~\cite{HR06}, the statement above in
terms of the sequence~$a_n$ is necessary, and is, for instance, also present
when dealing with the maximum of $n$ i.i.d.\ geometric random variables.

Examples of random fields for which Theorem~\ref{thm:Gumbel} applies are given
in the following corollary:

\begin{corollary}{\bf(Examples subcritical Gumbel maximal clusters)}
\label{cor-examplesdubGumbel}
	The conclusions in Theorem~\ref{thm:Gumbel} hold in the following special
	cases:
	\begin{enumerate}
		\item[(i)] Subcritical percolation;
		\item[(ii)] Subcritical Ising model;
		\item[(iii)] Subcritical random-cluster models satisfying the
		FKG-property, for which $\mu<1$.
	\end{enumerate}
\end{corollary}

The assumed FKG-property in Corollary~\ref{cor-examplesdubGumbel}(iii) is
necessary to ensure that~\eqref{secmcon} applies. See e.g.~\cite{Grim06} for a
discussion of the FKG-property for random-cluster measures, and \cite[Theorems
5.55, 5.86]{Grim06} for examples of parameter values for which the
random-cluster measure satisfies $\mu<1$.

Theorem~\ref{thm:Gumbel} follows from \cite[Theorem~3.6]{HR06}, and, in the
case of percolation, from \cite[Theorem~1.1]{HR06} combined with
\cite[Theorem~1.5]{HR06}. We note that in \cite{HR06}, it was assumed that the
measure $\Prob$ has so-called \emph{subcritical clusters}\footnote{We take
this opportunity to correct a mistake in Definition~3.3(i) in~\cite{HR06}: the
condition that $\Prob( |\Cle(0)| < \infty )=1$ should read $\Prob( |C(0)| <
\infty )=1$. Similarly in Definition~3.3(ii).} \cite[Definition~3.3(i)]{HR06},
which is implied by the ratio limit theorem with $\mu<1$,
Corollary~\ref{cor:ratioCle} above.  In fact, Corollary~\ref{cor:ratioCle}
implies that $\xi$ and~$\zeta$, defined in \cite[Definition~3.3(i)]{HR06}, are
equal so that the strongest version in \cite[Theorem~3.6]{HR06} applies.

We now turn to the maximal supercritical cluster, which was investigated
in~\cite{HR06} only in the context of site percolation, to which we will
therefore restrict ourselves here as well. For supercritical~$p$, we define
\begin{equation}
\label{meenzbsup}
	|\Cmax| = \max_{x\in B_n: |C(x)| < \infty} |C(x)|
\end{equation}
to be the largest \emph{finite} cluster intersecting the cube. The ratio limit
theorem implies that, in the language of \cite[Definition~3.3(ii)]{HR06},
percolation has supercritical clusters. Therefore, \cite[Theorem~3.9]{HR06}
implies the following Gumbel statistics for the largest finite supercritical
cluster:

\begin{theorem}{\bf(Supercritical Gumbel maximal cluster for percolation)}
\label{thm:Gumbelsup}
	Let $p_c<p<1$ and let $\Prob_p$ denote the percolation measure with
	parameter~$p$. There exists a sequence~$u_n(x)$ with $u_n(x)\in\N$ and
	$u_n(x)\to\infty$ for all $x\in\R$ as $n\to\infty$, such that for all
	$x\in\R$,
	\[
		\Prob_p(|\Cmax| \leq u_n(x)) = e^{-e^{-x}} + o(1),
	\]
	where the error term may depend on~$x$.
\end{theorem}

\subsection{Further notation used in the proofs}
\label{ssec:notation}

Throughout the rest of the paper, we will write $c_n$ for the probability that
the cluster of the origin has size~$n$, and $p_n$ for the probability that the
cluster of the origin is finite and has size at least~$n$. That is, $c_n :=
\Prob( |C|=n )$ and $p_n := \Prob( n\leq|C|<\infty ) = \sum_{m=n}^\infty c_m$.

\section{Proof of the pattern theorem}
\label{sec:pattern}

This section is devoted to the proof of the pattern theorem,
Theorem~\ref{thm:pattern}. Our proof is similar to the proof in~\cite{Madr99}
of a pattern theorem stated in a different context. We recall that the pattern
theorem states that for a given pattern~$P$, there exists an $a>0$ such that
it is very unlikely that $P$ occurs at less than $an$ distinct sites of the
grid $V = (r+2)\Z^d + (1,1,\ldots,1)$ on a cluster of size~$n$.

The proof proceeds roughly as follows. If we assume that one is likely to see
at most $an$ occurrences of~$P$ on~$C$ at sites of the grid~$V$, then there
are many sites left on this grid where we can create new occurrences of~$P$
on~$C$. By creating a single new occurrence, we may change the size of~$C$,
but never by  more than $|\overline{Q}| = (r+2)^d$ sites. Consider all
configurations we can generate by introducing $\delta n$ new occurrences of
$P$ on~$C$. For every configuration we start from, there is an exponential
number of ways to introduce $\delta n$ occurrences of $P$ on~$C$, and all
generated configurations contribute to $c_m$ for some $m \in [n - |Q|\delta n,
n + |Q|\delta n]$, and hence to $p_{n - |Q| \delta n}$ in particular. However,
introducing occurrences of~$P$ will change the probability, and it is
furthermore clear that many of the generated configurations will be obtained
multiple times from different starting configurations. Still, if the random
field has the finite-energy property and $\delta$ is small enough, we will
generate so many distinct configurations that this contribution wins, and this
will prove Theorem~\ref{thm:pattern}. Now let us fill in the details of the
proof.

\begin{proof}[Proof of Theorem~\ref{thm:pattern}]
	We want to study the probability that the cluster of the origin has
	size~$n$, and $P$ occurs on~$C$ at no more than~$an$ distinct sites
	of~$V$. So, for all $n \geq 1$ define the set of relevant configurations
	by
	\begin{equation}
		S_n = \{ \omega \in \Omega: |C|=n, N_P \leq an \}.
	\end{equation}
	Consider a configuration $\omega \in S_n$. Then~$C$ has size~$n$, which
	implies that there are at least $n/|\overline{Q}| =: bn$ extended cubes at
	sites of~$V$ that intersect~$C$. No more than~$an$ of these cubes can
	contain an occurrence of~$P$. Thus, assuming $a<b$, with every
	configuration $\omega \in S_n$ we can associate a collection $X(\omega)
	\subset V$ of exactly $bn-an$ sites such that the extended cubes at these
	sites intersect~$C$, and $P$ does not occur at any of these sites.

	For any configuration $\omega \in S_n$ and site $x \in X(\omega)$, we can
	create an occurrence of~$P$ at~$x$ on~$C$ by vacating all occupied sites
	of $\partial \overline{Q}_x$ that do not belong to~$C$, occupying all
	sites of~$\partial Q_x$, and changing the configuration inside~$Q_x$ into
	an occurrence of~$P$ at~$x$. Here we emphasise that we first vacate sites
	on the boundary of~$\overline{Q}_x$ to avoid connecting the occupied
	cluster of the origin to another occupied cluster, whose size we cannot
	control and might be infinite. Also note that by the finite-energy
	property~\eqref{finiteE}, the factor by which the probability changes upon
	a single introduction of~$P$ is bounded from below by a constant of the
	form $\exp(-A \, |\partial \overline{Q} \cup \overline{Q}|) = \exp\bigl(
	-A \, (r+4)^d \bigr)$ for some fixed $A>0$.

	Now, for every $\omega \in S_n$, consider  all possible ways of creating
	occurrences of~$P$ at exactly $\delta n$ sites chosen from the
	collection~$X( \omega )$, where $\delta > 0$ is a small number that will
	be fixed later on. Write~$S'_n$ for the collection of all configurations
	generated in this way. Then in general, the same configuration $\omega'
	\in S'_n$ can be obtained from multiple configurations in~$S_n$. However,
	the only differences between these configurations can be the local
	configurations inside the extended cubes and their boundaries at those
	sites of~$V$ where~$P$ occurs on the cluster~$C(\omega')$. One can have
	$q^{| \overline{Q} \cup \partial \overline{Q} |}$ different configurations
	inside an extended cube plus its boundary, where we recall that~$q$ is the
	size of the state space~$S$ per site. Also, there are at most
	$(\delta+a)n$ occurrences of~$P$ on~$C$ if we create $\delta n$ new
	occurrences.  Therefore, on the one hand,
	\begin{equation}
	\label{pattern1}
		\Prob(S'_n) \geq \binom{bn-an}{\delta n}
		e^{-A \, |\overline{Q} \cup \partial \overline{Q}| \, \delta n} \,
		q^{-(\delta+a)n \, |\overline{Q} \cup \partial \overline{Q}|} \,
		\Prob(S_n).
	\end{equation}
	On the other hand it is clear that for any $\omega' \in S'_n$, the
	occupied cluster of the origin is finite and has size at least $n - |Q| \,
	\delta n$. Therefore,
	\begin{equation}
	\label{pattern2}
		\Prob(S'_n) \leq p_{n - |Q| \, \delta n}.
	\end{equation}

	We now divide \eqref{pattern1} and~\eqref{pattern2} by~$c_n$, combine the
	two resulting inequalities, take $n$-th roots on both sides and then the
	$\limsup_{n \to \infty}$, using~\eqref{limit} and Stirling's formula. This
	leads to
	\begin{multline}
		\mu^{-|Q| \, \delta} \geq
		\frac{(b-a)^{b-a}}{\delta^\delta \, (b-a-\delta)^{b-a-\delta}} \,
		e^{-A \, |\overline{Q} \cup \partial \overline{Q}| \, \delta} \,
		q^{-(\delta+a) \, |\overline{Q} \cup \partial \overline{Q}|} \\
		\null \times \limsup_{n\to\infty}
		\left[ \Prob_n(N_P \leq an) \right]^{1/n}.
	\end{multline}
	At this point we may just as well take $a=\delta$. Setting $t=\delta /
	(b-a)$ and $\tilde{\mu} = \mu^{-|Q|} \, e^{A |\overline{Q} \cup \partial
	\overline{Q}|} \, q^{2|\overline{Q} \cup \partial \overline{Q}|}$, the
	previous inequality can be rewritten as
	\begin{equation}
		\limsup_{n\to\infty} \left[ \Prob_n(N_P \leq \delta n) \right]^{1/n}
		\leq \left( t^t \, (1-t)^{1-t} \, \tilde{\mu}^t \right)^{b-\delta}.
	\end{equation}
	The right-hand side is smaller than~$1$ whenever $t = \delta / (b-a) <
	\tilde{\mu}^{-1}$.  Therefore, the left-hand side is smaller than~$1$ for
	sufficiently small $\delta>0$, which proves Theorem~\ref{thm:pattern}.
\end{proof}

\section{Proofs of the two-pattern theorems}
\label{sec:twopattern}

In this section we are interested in the ratio between the numbers of
occurrences of two distinct pattern $P$ and $P'$ on~$C$. We will prove, as
stated in Theorems \ref{thm:twopattern} and~\ref{thm:strongtwopattern}, that
if the random field has the boundary-$1$ Markov property, then the ratio
$N_P/N_{P'}$ must be close to a fixed number $\gamma = \gamma_{PP'}$ defined
in~\eqref{gamma}. The basic strategy of the proofs is as follows. We will
consider those configurations such that $N_P-\gamma N_{P'}$ differs from~$0$
by at least $\epsilon n^\alpha$ occurrences for some $\epsilon>0$ and
$0<\alpha\leq1$. Then we can make this difference smaller by either turning
occurrences of~$P$ into occurrences of~$P'$ or the other way around. By
deriving a bound on the probability of the collection of configurations
generated in this way, in a similar way as in the proof of
Theorem~\ref{thm:pattern}, we can then show that the probability that $|N_P -
\gamma N_{P'}|$ is at least $\epsilon n^\alpha$ must be small. We start by
proving Theorem~\ref{thm:twopattern}, in which $\alpha=1$, after which we
prove Theorem~\ref{thm:strongtwopattern}.

\begin{proof}[Proof of Theorem~\ref{thm:twopattern}]
	We will show that under the conditions stated in the theorem, for every
	choice of $P$ and~$P'$ (and every possible value of the corresponding
	$\gamma = \gamma_{PP'}$), and for all $\epsilon>0$,
	\begin{equation}
	\label{twopattern1}
		\limsup_{n\to\infty}
		\big[ \Prob_n( N_P \leq \gamma N_{P'} - \epsilon n) \big]^{1/n}
		\leq \frac{ \bigl( 1+\frac{\epsilon}{1+\gamma}
					\bigr)^{1+(1+\epsilon) \, \gamma^{-1}}}
				{(1+\epsilon)^{(1+\epsilon) \, \gamma^{-1}}}
		< 1.
	\end{equation}
	Since this holds for any choice of the two patterns, we can interchange
	the roles of $P$ and~$P'$ in~\eqref{twopattern1}, which also replaces
	$\gamma = \gamma_{PP'}$ by $\gamma^{-1} = \gamma_{P'P}$ and $\epsilon$ by
	$\epsilon \gamma^{-1}$, to obtain
	\begin{equation}
	\label{twopattern2}
		\limsup_{n\to\infty}
		\big[ \Prob_n( N_P \geq \gamma N_{P'} + \epsilon n) \big]^{1/n}
		\leq \frac{ \bigl( 1+\frac{\epsilon}{1+\gamma}
					\bigr)^{1+\gamma+\epsilon}}
				{\bigl( 1+\frac{\epsilon}{\gamma} \bigr)^{\gamma+\epsilon}}
		< 1.
	\end{equation}
	These two results together evidently imply Theorem~\ref{thm:twopattern}.
	Hence, it suffices to derive~\eqref{twopattern1}.

	To derive~\eqref{twopattern1}, let $S_n$ be the collection of
	configurations such that $|C|=n$ and $N_P \leq \gamma N_{P'} - \epsilon
	n$. For every $\omega \in S_n$, consider all possible ways in which we can
	change $\delta \, \epsilon n$ occurrences of $P'$ on~$C$ at sites of~$V$
	into occurrences of $P$ on~$C$, where $0 < \delta < 1$ will be fixed later
	on. Then, for given $N_P$ and $N_{P'}$, the ratio of the number of
	configurations generated in this way to the number of configurations from
	which they were obtained, is given by the factor
	\begin{equation}
	\begin{split}
	\label{twopatternfactor}
		&\binom{N_P + N_{P'}}{N_P + \delta \, \epsilon n} \,
		\binom{N_P + N_{P'}}{N_P}^{-1} \\
		&\qquad\qquad = \frac{N_P!}{(N_P + \delta \, \epsilon n)!} \,
		\frac{N_{P'}!}{(N_{P'} - \delta \, \epsilon n)!} \\
		&\qquad\qquad \geq \frac{N_P!}{(N_P + \delta \, \epsilon n)!} \,
		\frac{(\gamma^{-1} N_P + \gamma^{-1} \, \epsilon n)!}{(\gamma^{-1} N_P
		+ (\gamma^{-1}-\delta) \, \epsilon n)!},
	\end{split}
	\end{equation}
	where the last inequality follows from $N_P \leq \gamma N_{P'} - \epsilon
	n$.

	Now we observe (by differentiating with respect to~$N$) that for all
	$k=1,2,\ldots,\delta \, \epsilon n$, the fraction $(\gamma^{-1}N +
	(\gamma^{-1} - \delta) \, \epsilon n + k) / (N + k)$ is non-increasing
	in~$N$. Therefore, using the fact that $N_P$ is necessarily smaller
	than~$n$, we can bound the factor~\eqref{twopatternfactor} by
	\begin{equation}
	\begin{split}
		&\prod_{k=1}^{\delta \, \epsilon n}
		\frac{\gamma^{-1}N_P + (\gamma^{-1} - \delta) \, \epsilon n + k}
			{N_P + k} \\
		&\qquad\qquad \geq \prod_{k=1}^{\delta \, \epsilon n}
		\frac{\gamma^{-1}n + (\gamma^{-1} - \delta) \, \epsilon n + k}
			{n + k} \\
		&\qquad\qquad = \frac{n!}{(n + \delta \, \epsilon n)!} \,
		\frac{(\gamma^{-1} n + \gamma^{-1} \, \epsilon n)!}
			{(\gamma^{-1} n + (\gamma^{-1}-\delta) \, \epsilon n)!}.
	\end{split}
	\end{equation}

	For every configuration obtained from $\omega\in S_n$ by changing $\delta
	\, \epsilon n$ occurrences of $P'$ on~$C$ into occurrences of $P$ on~$C$,
	the cluster of the origin has size $|C| = n - \delta \, \epsilon n \,
	\delta_c$, where $\delta_c = c_{P'} - c_P$. Moreover, by virtue of the
	boundary-$1$ Markov property, every change of $P'$ on~$C$ into~$P$ changes
	the probability of a configuration by the factor $\Prob(\Box P) /
	\Prob(\Box P')$, where we recall the definition~\eqref{PBox} of
	$\Prob(\Box P)$. Therefore, we can write
	\begin{multline}
		\frac{c_{n - \delta \, \epsilon n \, \delta_c}}{c_n} \geq
		\left( \frac{\Prob(\Box P)}{\Prob(\Box P')}
		\right)^{\delta \, \epsilon n}
		\frac{n!}{(n + \delta \, \epsilon n)!} \,
		\frac{(\gamma^{-1} n + \gamma^{-1} \, \epsilon n)!}
			{(\gamma^{-1} n + (\gamma^{-1}-\delta) \, \epsilon n)!} \\
		\null \times \Prob_n(N_P \leq \gamma N_{P'} - \epsilon n).
	\end{multline}
	Taking the $n$-th root on both sides and then the $\limsup_{n\to\infty}$,
	using~\eqref{limit}, \eqref{gamma} and Stirling's formula, leads to
	\begin{equation}
	\label{twopatternf1}
		1 \geq f_\delta(\epsilon) \, \limsup_{n\to\infty}
		\left[ \Prob_n(N_P \leq \gamma N_{P'} - \epsilon n)
		\right]^{1/n},
	\end{equation}
	where
	\begin{equation}
		f_\delta(x) =
		\frac{(1+x)^{(1+x)\gamma^{-1}}}
		{(1 + (1-\delta\gamma) x)^{\gamma^{-1} + (\gamma^{-1}-\delta) x}
			\, (1 + \delta x)^{1 + \delta x}}.
	\end{equation}

	Observe that $f_\delta(0)=1$, and taking the derivative of $f_\delta(x)$
	with respect to~$x$ yields
	\begin{equation}
		\frac{f_\delta'(x)}{f_\delta(x)} = \frac{1}{\gamma} \,
		\log\left( \frac{1+x}{1+x-\delta\gamma x} \right)
		+ \delta \, \log\left( \frac{1+x-\delta\gamma x}{1+\delta x} \right).
	\end{equation}
	It follows that for all $x>0$ and $\delta \leq (1+\gamma)^{-1}$,
	$f_\delta'(x) > 0$ and therefore $f_\delta(x) > 1$. It is not difficult to
	see that for fixed $x>0$, $f_\delta(x)$ is actually maximal at $\delta =
	(1+\gamma)^{-1}$. We therefore set $\delta = (1+\gamma)^{-1}$, and then
	the desired result~\eqref{twopattern1} follows from~\eqref{twopatternf1}.
	This proves Theorem~\ref{thm:twopattern}.
\end{proof}

\paragraph{Remark.}
The bound appearing on the right-hand side of~\eqref{twopattern1} is not best
possible in general. Indeed, if we restrict $\epsilon$ to the range
$(0,\gamma)$, then the combinatorial factor~\eqref{twopatternfactor} can also
be bounded by
\begin{equation}
\begin{split}
	\frac{N_P!}{(N_P + \delta \, \epsilon n)!} \,
	\frac{N_{P'}!}{(N_{P'} - \delta \, \epsilon n)!}
	&\geq \frac{(\gamma N_{P'} - \epsilon n)!}
			{(\gamma N_{P'} - (1-\delta) \, \epsilon n)!} \,
		\frac{N_{P'}!}{(N_{P'} - \delta \, \epsilon n)!} \\
	&\geq \frac{(\gamma n - \epsilon n)!}
			{(\gamma n - (1-\delta) \, \epsilon n)!} \,
		\frac{n!}{(n - \delta \, \epsilon n)!}.
\end{split}
\end{equation}
Here, as before, the first inequality follows from $N_P \leq \gamma N_{P'} -
\epsilon n$ and the second is a consequence of the fact that $N_{P'}$ is
necessarily less than~$n$. The same reasoning as in the previous proof then
leads us again to an inequality of the form~\eqref{twopatternf1}, where now
the function $f_\delta(x)$ is given by
\begin{equation}
	f_\delta(x) =
	\frac{(1 - x\gamma^{-1})^{\gamma-x}}
		{(1 - (1-\delta) x\gamma^{-1})^{\gamma - (1-\delta) x} \,
			(1 - \delta x)^{1 - \delta x}}.
\end{equation}
As before, for fixed $0<x<\gamma$, this function is maximal and larger
than~$1$ at $\delta = (1+\gamma)^{-1}$. This leads for $0<\epsilon<\gamma$ to
the bound
\begin{equation}
	\limsup_{n\to\infty}
	\big[ \Prob_n(N_P \leq \gamma N_{P'} - \epsilon n) \big]^{1/n}
	\leq \frac{ \bigl( 1-\frac{\epsilon}{1+\gamma}
				\bigr)^{1+\gamma-\epsilon}}
			{\bigl( 1-\frac{\epsilon}{\gamma} \bigr)^{\gamma-\epsilon}}
	< 1.
\end{equation}
This bound is better than the bound in~\eqref{twopattern1} for small values
of~$\gamma$, but worse than~\eqref{twopattern1} for large values of~$\gamma$.

\begin{proof}[Proof of Theorem~\ref{thm:strongtwopattern}]
	The proof of Theorem~\ref{thm:strongtwopattern} is similar to the proof of
	Theorem~\ref{thm:twopattern} above. We define the set of relevant
	configurations by
	\begin{equation}
		S_n = \{ \omega \in \Omega : |C| = n, N_P - \gamma N_{P'}  \geq
		\epsilon n^\alpha \}.
	\end{equation}
	For every $\omega \in S_n$, we can choose $\delta \, \epsilon n^\alpha$
	occurrences of $P$ on~$C$ at sites of~$V$ and turn them into occurrences
	of~$P'$, where $0 < \delta < 1$ is a number that we will fix later. This
	will change the size of~$C$ to $n + \delta \, \epsilon n^\alpha \,
	\delta_c$, since $\delta_c = c_{P'} - c_P$ is the change in~$|C|$ if we
	replace a single occurrence of $P$ on~$C$ by an occurrence of~$P'$.

	For given $N_P$ and $N_{P'}$, the ratio of the number of generated
	configurations in which there are $N_P - \delta \, \epsilon n^\alpha$
	occurrences of $P$ on~$C$ at sites of~$V$ and $N_{P'} + \delta \, \epsilon
	n^\alpha$ occurrences of $P'$ on~$C$ at sites of~$V$ to the number of
	configurations in~$S_n$ from which they can be generated, is, similarly
	to~\eqref{twopatternfactor}, given by
	\begin{equation}
	\begin{split}
		&\binom{ N_P+N_{P'} }{ N_P - \delta \, \epsilon n^\alpha }
		\binom{ N_P+N_{P'} }{ N_P }^{-1} \\
		&\qquad\qquad = \frac{N_P!}{(N_P - \delta \, \epsilon n^\alpha)!} \,
			\frac{N_{P'}!}{(N_{P'} + \delta \, \epsilon n^\alpha)} \\
		&\qquad\qquad \geq \frac{(\gamma N_{P'} + \epsilon n^\alpha)!}
				{(\gamma N_{P'} + (1-\delta) \epsilon n^\alpha)!} \,
			\frac{N_{P'}!}{(N_{P'} + \delta \, \epsilon n^\alpha)!} \\
		&\qquad\qquad \geq \frac{(\gamma n + \epsilon n^\alpha)!}
				{(n - \delta \, \epsilon n^\alpha)!} \,
			\frac{n!}{(n + \delta \, \epsilon n^\alpha)!}.
	\end{split}
	\end{equation}
	Here, the first inequality follows from $N_P - \gamma N_{P'} \geq \epsilon
	n^\alpha$, and in the second inequality we used monotonicity of the
	expression in~$N_{P'}$, which can be shown by an argument similar to the
	one below~\eqref{twopatternfactor}, together with the fact that $N_{P'}
	\leq n$. We note that we have assumed that $0 < \delta \leq \gamma^{-1}$
	to obtain the second inequality.

	Because for every configuration generated from $\omega\in S_n$, we have
	that $|C| = n+\delta \, \epsilon n^\alpha \, \delta_c$, we can write
	\begin{multline}
		\frac{c_{n + \delta \, \epsilon n^\alpha \, \delta_c}}{c_n}
		\geq
		\left( \frac{\Prob(\Box P')}{\Prob(\Box P)}
		\right)^{\delta\,\epsilon n^\alpha}
		\frac{(\gamma n + \epsilon n^\alpha)!}
			{(n - \delta \, \epsilon n^\alpha)!} \,
		\frac{n!}{(n + \delta \, \epsilon n^\alpha)!} \\
		\null \times \Prob_n(N_P-\gamma N_{P'} \geq \epsilon n^\alpha).
	\end{multline}
	Using Stirling's formula and substituting~\eqref{gamma} for $\gamma =
	\gamma_{PP'}$, this expression can be rewritten as
	\begin{multline}
		\frac{c_{n + \delta \, \epsilon n^\alpha \, \delta_c}}{c_n}
		\geq \mu^{\delta \, \epsilon n^\alpha \, \delta_c} \,
		e^{(\gamma^{-1}\delta - \ahalf\gamma^{-1}\delta^2 - \ahalf\delta^2)
			\epsilon n^{2\alpha - 1} + o(n^{2\alpha - 1})} \\
		\null \times \Prob_n(N_P-\gamma N_{P'} \geq \epsilon n^\alpha).
	\end{multline}
	We now choose $\delta = (1 + \gamma)^{-1}$, since this maximizes the
	middle stretched exponential term on the right-hand side. Raising to the
	power $n^{1-2\alpha}$ (here we assume $\alpha > \ahalf$) and taking the
	$\limsup_{n\to\infty}$ then leads to
	\begin{multline}
	\label{strongtwopattern}
		e^{-\ahalf(\gamma+\gamma^2)^{-1} \, \epsilon} \,
		\limsup_{n\to\infty}
		\left[
			\frac{c_{n + \delta \, \epsilon n^\alpha \, \delta_c}}{c_n} \,
			\mu^{-\delta \, \epsilon n^\alpha \, \delta_c}
		\right]^{1/n^{2\alpha-1}} \\
		\geq
		\limsup_{n\to\infty}
		\left[
			\Prob_n(N_P - \gamma N_{P'} \geq \epsilon n^\alpha)
		\right]^{1/n^{2\alpha-1}}.
	\end{multline}
	Observe that the left-hand side in~\eqref{strongtwopattern} is smaller
	than~$1$ if $\delta_c = 0$,  which immediately proves the first claim of
	the theorem.

	For $\delta_c \neq 0$ we need a bound on the ratio of cluster-size
	probabilities appearing on the left-hand side of~\eqref{strongtwopattern}.
	The exponential decay~\eqref{limit} does not lead to useful bounds, but in
	the supercritical case, where $\mu=1$, the stretched exponential
	decay~\eqref{superlimit} tells us that
	\begin{equation}
		\limsup_{n\to\infty} \left[ \frac{c_{n + \delta \, \epsilon n^\alpha
		\, \delta_c}}{c_n} \right]^{1/n^\beta} = 1.
	\end{equation}
	In~\eqref{strongtwopattern}, this makes the left-hand side smaller
	than~$1$ if we take $2\alpha - 1 \geq \beta$, that is, $\alpha \geq \ahalf
	(1+\beta)$. This completes the proof of
	Theorem~\ref{thm:strongtwopattern}.
\end{proof}

To conclude this section, we strengthen the above bound in the case $\delta_c
< 0$ under the condition~\eqref{supermulti} of supermultiplicativity. This
gives
\begin{equation}
	\frac{c_{n - \delta \, \epsilon n^\alpha \, |\delta_c|}}{c_n} \leq \frac1A
	\, \frac{\delta \, \epsilon n^\alpha \, |\delta_c|}{ c_{\delta \, \epsilon
	n^\alpha \, |\delta_c|} }.
\end{equation}
Using the stretched exponential decay~\eqref{superlimit}, we obtain from this
inequality
\begin{equation}
	\limsup_{n\to\infty} \left[ \frac{ c_{n - \delta \, \epsilon n^\alpha \,
	|\delta_c|} }{ c_n } \right]^{1/n^{\alpha\beta}} \leq e^{\xi (\delta \,
	\epsilon \, |\delta_c|)^{\beta} }
\end{equation}
for some $0 < \xi < \infty$. We note that $\alpha\beta = 2\alpha-1$ if we take
$\alpha = (2-\beta)^{-1}$. Then, if we insert the previous expression
into~\eqref{strongtwopattern}, we see that the left-hand side becomes smaller
than~$1$ if we take $\epsilon$ larger than some $a_0 > 0$, which proves the
statement in the Remark following Theorem~\ref{thm:strongtwopattern}.

\section{Proofs of the ratio limit theorems}
\label{sec:ratio}

We shall now show how the pattern theorems proved above can be combined to
derive the ratio limit theorems (Corollaries \ref{cor:ratio}
and~\ref{cor:strongratio}). For this, we shall take a fixed cube diameter
$r=3$, and consider two specific patterns $P$ and~$P'$: $P$ is the pattern
such that the site $(1,1,\ldots,1)$ is occupied and all other sites of~$Q$ are
vacant, and $P'$ is the pattern such that the origin is occupied and all other
sites of~$Q$ are vacant, see Figure~\ref{fig:patterns}.  For these two
patterns and integers $i,j\geq 0$, we introduce the notation
\begin{equation}
\label{c_n(i,j)}
	c_n(i,j) = \Prob_p(|C|=n, N_P = i, N_{P'} = j).
\end{equation}
Observe that in our earlier notation used to formulate the pattern theorems,
$\delta_c = c_{P'} - c_P = 1$. For percolation, the patterns $P$ and $P'$ are
chosen such that whenever we change an occurrence of $P$ on~$C$ into an
occurrence of $P'$ on~$C$, we do not change the probability of the
configuration. This is, however, not generally the case for a Markovian random
field.

\begin{figure}
	\centering
	\begin{picture}(200,66)
		\put(0,0){\includegraphics{Patterns}}
		\put(23,2){\makebox(0,0)[b]{\smash{$P$}}}
		\put(177,2){\makebox(0,0)[b]{\smash{$P'$}}}
	\end{picture}
	\caption{\label{fig:patterns}The patterns $P$ and~$P'$ in dimension $d=2$.
	White sites are vacant, black sites are occupied.}
\end{figure}

Our proofs of the ratio limit theorems are based on the following observation.
Let $S_n(i,j)$ be the collection of configurations such that $|C|=n$, $N_P =
i$ and $N_{P'} = j$. Then, for every $\omega \in S_n(i,j)$, there are $i$~ways
of changing one occurrence of $P$ on~$C$ at a site of~$V$ into an occurrence
of $P'$ on~$C$, leading to an $\omega' \in S_{n+1}(i-1, j+1)$. For each
configuration that we generate in this way, there are~$j$ other configurations
in the set $S_n(i,j)$ from which we could have obtained the same configuration
by changing one occurrence of $P$ on~$C$ at a site of~$V$ into an occurrence
of $P'$ on~$C$.  Therefore, for $i\geq1$ and $j\geq0$,
\begin{equation}
\label{changePtoP'}
	c_{n+1}(i-1,j+1) = \frac{i}{j+1} \, \frac{\Prob(\Box P')}{\Prob(\Box P)}
	\, c_n(i,j),
\end{equation}
where we have again used the boundary-$1$ Markov property. From this equality
one sees that bounds on the ratio $N_P / N_{P'}$ will lead directly to bounds
on $c_{n+1} / c_n$. We will now use this to prove Corollary~\ref{cor:ratio}
for random fields satisfying the boundary-$1$ Markov property.

\begin{proof}[Proof of Corollaries \ref{cor:ratio} and~\ref{cor:ratioCle}]
	First let us show that if $c_{n+1} / c_n$ converges to~$\mu$ and $p_n =
	\sum_{m=n}^\infty c_m$, then $p_{n+1} / p_n$ converges to~$\mu$ as well.
	To show this, fix $0 < \epsilon < \mu$. Since $c_{n+1} / c_n$ converges
	to~$\mu$, there exists an integer $N_\epsilon > 0$ such that
	\begin{equation}
		\left| \frac{c_{n+1}}{c_n}-\mu \right| \leq \epsilon
	\end{equation}
	for all $n > N_\epsilon$. Observing that
	\begin{equation}
		\frac{c_n}{p_n} = \left[ \sum_{m=0}^\infty \frac{c_{n+m}}{c_n}
		\right]^{-1} = \left[ 1 + \sum_{m=1}^\infty \prod_{k=1}^m
		\frac{c_{n+k}}{c_{n+k-1}} \right]^{-1},
	\end{equation}
	it follows that for $n > N_\epsilon$, if $\mu < 1$,
	\begin{align}
	\label{bound1}
		\frac{c_n}{p_n} &\leq \left[ \sum_{m=0}^\infty (\mu-\epsilon)^m
		\right]^{-1} = 1-\mu+\epsilon, \\
		\frac{c_n}{p_n} &\geq \left[ \sum_{m=0}^\infty (\mu+\epsilon)^m
		\right]^{-1} = 1-\mu-\epsilon.
	\end{align}
	Since $p_{n+1}/p_n = 1-c_n/p_n$, this implies $\lim_{n \to \infty} p_{n+1}
	/ p_n = \mu$ for every $0 < \mu < 1$. Now note that \eqref{bound1} also
	applies if $\mu=1$. Since $c_n/p_n \geq 0$ holds trivially, we also obtain
	$\lim_{n \to \infty} p_{n+1} / p_n = \mu$ when $\mu=1$. Thus, to prove
	Corollary~\ref{cor:ratio}, it remains to show that $c_{n+1} /c_n$
	converges to~$\mu$.

	Now let us write $c^\ast_n = \Prob\bigl( |\Cle(0)| = n \bigr)$ and
	$p^\ast_n = \sum_{m=n}^\infty c^\ast_m$. We can repeat the argument above
	(with $c^\ast_n, p^\ast_n$ in place of $c_n, p_n$) to see that
	$c^\ast_{n+1} / c^\ast_n \to\mu$ implies $p^\ast_{n+1} / p^\ast_n \to\mu$.
	However, for translation-invariant~$\Prob$, we have that $c_n = n \,
	c^\ast_n$ for all $n\geq1$ (this is Lemma~4.1 in~\cite{HR06}), so that
	$c^\ast_{n+1} / c^\ast_n \to\mu$ is implied by $c_{n+1} / c_n \to\mu$. We
	conclude that establishing that $c_{n+1} / c_n$ converges to~$\mu$
	suffices to prove not only Corollary~\ref{cor:ratio} but also
	Corollary~\ref{cor:ratioCle}.

	We will show that $c_{n+1} / c_n$ converges to~$\mu$ for a random field
	satisfying the boundary-$1$ Markov property by using Theorems
	\ref{thm:pattern} and~\ref{thm:twopattern}. Let the patterns $P$ and~$P'$
	be as above, and let $a>0$ be the constant appearing in
	Theorem~\ref{thm:pattern} for the pattern~$P'$. Using the
	notation~\eqref{c_n(i,j)} introduced above, we can write
	\begin{equation}
	\label{ratioeq}
		\frac{c_{n+1}}{c_n} = \sum_{j = -1}^n \sum_{i = 1}^n
		\frac{c_{n+1}(i-1,j+1)}{c_n}.
	\end{equation}
	We may use Theorems \ref{thm:pattern} and~\ref{thm:twopattern} together
	with~\eqref{limit}, to restrict the sums in~\eqref{ratioeq} at the cost of
	introducing an exponentially small error term. For convenience, we write
	$\gamma = \gamma_{PP'}$ for the constant defined in~\eqref{gamma}.
	Applying our observation~\eqref{changePtoP'}, we obtain
	\begin{equation}
	\begin{split}
		\frac{c_{n+1}}{c_n}
		&= \sum_{j = an}^n \,
			\sum_{i = \gamma j - o(n)}^{\gamma j + o(n)}
			\frac{c_{n+1}(i-1,j+1)}{c_n} + o(1) \\
		&= \sum_{j = an}^n \,
			\sum_{i = \gamma j - o(n)}^{\gamma j + o(n)}
			\frac{i}{j+1} \, \frac{\Prob(\Box P')}{\Prob(\Box P)} \,
			\frac{c_n(i,j)}{c_n} + o(1) \\
		&= \sum_{j = an}^n \,
			\sum_{i = \gamma j - o(n)}^{\gamma j + o(n)}
			\big(\gamma + o(1)\big) \, \frac{\Prob(\Box P')}{\Prob(\Box P)} \,
			\frac{c_n(i,j)}{c_n} + o(1) \\
		&= \big(\mu + o(1)\big) \sum_{j = 0}^n \, \sum_{i = 0}^n
			\frac{c_n(i,j)}{c_n} + o(1) \\
		&= \mu+o(1).
	\end{split}
	\end{equation}
	This proves that the ratio $c_{n+1} / c_n$ converges to~$\mu$, which
	completes the proof of Corollaries \ref{cor:ratio}
	and~\ref{cor:ratioCle}.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:strongratio}]
	Using the same notations as above, by Theorem~\ref{thm:strongtwopattern}
	we can write
	\begin{equation}
	\label{strongratioproof}
		\frac{c_{n+1}}{c_n} = \sum_{j = an}^n \, {\sum_i}' \frac{i}{j+1} \,
		\frac{\Prob(\Box P')}{\Prob(\Box P)} \,
		\frac{c_n(i,j)}{c_n} + o\bigl( n^{-(1-\beta)/2} \bigr),
	\end{equation}
	where the prime on the second sum means that $i$ is restricted to run from
	$\gamma j - \epsilon n^{(1+\beta)/2}$ to $\gamma j + \epsilon
	n^{(1+\beta)/2}$, with arbitrary $\epsilon>0$. Proceeding as in the proof
	of Corollary~\ref{cor:ratio}, this implies that
	\begin{equation}
		\left| \frac{c_{n+1}}{c_n} - 1 \right| \leq
		\frac{2\epsilon}{an^{(1-\beta)/2}}
	\end{equation}
	for sufficiently large~$n$, as required. This is turn implies
	\begin{equation}
		\frac{c_{n+\lfloor xn^\alpha \rfloor}}{c_n}
		= \prod_{k=1}^{\lfloor xn^\alpha \rfloor} \frac{c_{n+k}}{c_{n+k-1}}
		= \left[ 1 + o\bigl( n^{-(1-\beta)/2} \bigr)
			\right]^{\lfloor xn^\alpha \rfloor}
		= 1+o(1)
	\end{equation}
	for all $x>0$ and $0 < \alpha \leq (1-\beta)/2$, establishing
	Corollary~\ref{cor:strongratio}.
\end{proof}

We close off this section by proving the statement in the Remark below
Corollary~\ref{cor:strongratio}. By the Remark following
Theorem~\ref{thm:strongtwopattern}, under~\eqref{supermulti} we may
restrict~$i$ in the primed sum in~\eqref{strongratioproof} to the range
$[\gamma j - a_0 n^{1/(2-\beta)},n]$. The same reasoning as before then leads
to the desired result
\begin{equation}
	\frac{c_{n+1}}{c_n} \geq 1 - \frac{2a_0}{an^{(1-\beta)/(2-\beta)}}.
\end{equation}

\paragraph{Acknowledgements.}
The work of RvdH was supported in part by the Netherlands Organization for
Scientific Research (NWO). We thank Geoffrey Grimmett for clarifying the proof
of~\eqref{limit} in the boundary-$0$ Markov case, as well as for pointing us
to \cite{Alex98, Alex04} on mixing conditions.

redpajama_set_name: RedPajamaArXiv


this frosted funfetti layer cake. dying.
this violet syrup and soda. for real?!
this burnt romaine with grilled avocado, tomatoes and cotija cheese. just stop.
this bourbon pecan pie with dark chocolate. can't even.
this hazelnut praline ice cream. seriousness.
this DIY wallpaper with a sharpie. say what?
this perfect recipe for a sliders party. gimme all the burgers.
this perfect porchetta recipe. mouth watering.
these toasted oatmeal chocolate chip cookies. yes please.
this mango margarita with chili salt and lime. bottoms up.
this vegan carrot cake with macadamia frosting. whoa.
these bbq beef beet burgers with avocado goat cheese. just shut it.
this rhubarb panna cotta tart. so pretty.
this spring quinoa with honey lemon vinaigrette. perfection.
this avocado tartare. that's my kind of tartare.
this 6 minute creamy tomato soup. craving.
these chocolate chip s'mores cookie sandwiches. holy.
Soba Noodle Salad with Caramelized Chicken and Chili Oil.
That bourbon pecan pie had my mouth watering. Looks so good!
I didn't even realize that avocado tartare was something I needed so badly in my life until now.
Yay for spring! I always love your lists!
I love that Frosted Funfetti Layer Cake. I wish I had a big slice right now.
Just ordered your cookbook! Can't wait for it to ship. Congratulations!
You are the kindest of the kind – and HOT DAYUM, those s'mores cookie sandwiches? I will take 1000.
I am heading over to those smores cookies. Can it get any better than that??
ohh that quinoa salad looks delicious! and love the funfetti cake !
Jessica!!! I've been a fan of your blog for a while now and was so excited to see that you've been crushing on some of my recipes ;-) Thanks so much for the link love!
That bourbon pecan pie with dark chocolate sounds amazing!!
where is the above beach photo from? thanks.
Great list. Some will be added to my query list :D i haven't a lot of time for a cooking. HOWSWEET thanks for a awesome blog.


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(H) 03/27/19 6:00 PM vs Wyomissing Area * Gymnasium (Geigle Complex-High School Gym) .
(A) 03/28/19 6:00 PM vs Wilson HS .
(A) 04/02/19 6:00 PM vs Exeter Township Senior HS .
(H) 04/04/19 6:00 PM vs Fleetwood Area * Gymnasium (Geigle Complex-High School Gym) .
(H) 04/09/19 6:00 PM vs Daniel Boone Area HS * Gymnasium (Geigle Complex-High School Gym) .
(A) 04/11/19 6:00 PM vs Governor Mifflin HS .
(H) 04/16/19 6:00 PM vs Wilson HS * Gymnasium (Geigle Complex-High School Gym) .
(A) 04/18/19 5:30 PM vs Central Dauphin East HS .
(H) 04/23/19 6:00 PM vs Exeter Township Senior HS * Gymnasium (Geigle Complex-High School Gym) .
(H) 04/24/19 6:00 PM vs Berks Catholic HS * Gymnasium (Geigle Complex-High School Gym) .
(A) 04/25/19 6:00 PM vs Fleetwood Area .
(A) 04/29/19 6:00 PM vs Conestoga Valley .
(A) 04/30/19 6:00 PM vs Daniel Boone Area HS .
(H) 05/02/19 6:00 PM vs Governor Mifflin HS * Gymnasium (Geigle Complex-High School Gym) .


text:
Thrill of the game under attack from endless ads
by Christina Ongley
Christina Ongley Senior Sub-editor
I'm the daughter of retired small businesspeople who has moved about the country a lot with my parents' ventures and, before that, my dad's army career. I love making a home out of wherever I am, and I have a passion for getting to know people and telling their stories. I've been a journalist for more than 15 years and I haven't met a person yet who doesn't have an interesting story to tell. When I'm not at work, you can find me hanging out with my son and trying to find some time to read a good...
DESPITE not really caring about the result of the final State of Origin match, like many other people I still got sucked into watching it last night.
I didn't really care whether it was a whitewash - or a "blue rinse", as some were calling it. The series had been decided regardless of the final match. I just wanted to watch a good game of football.
I'm a sports fan from way back, but I've been tiring of State of Origin and rugby league in general for many years, and this year was the first series in a while I've bothered to watch because I figured it might actually be a competition.
All in all, I did get a pretty good few games of football. But something has changed since I last tuned into Origin closely - because I also got a barrage of advertisements.
I'm not talking about the ads they play in between passages of play that we often wish we could fast-forward through. I'm talking about the ones that seemed to spill so effortlessly out of the mouths of the commentators.
I wasn't sure where the biggest competition was happening - on the field or behind the Channel Nine executives' doors, trying to figure out just how many product placements and sponsor messages they could fit into one commentary shift.
Years ago the State of Origin was marketed along the humorous lines of "cockroaches versus the cane toads" or the impassioned cry of "mate against mate, state against state".
Now the passion seems to have been replaced with an ad-packed Holden State of Origin. Yes, you heard Ray Warren say it a number of times: "The Holden State of Origin."
In case you missed it, you wouldn't have been under any illusions as to who the major sponsor was when you reached Holden Half-Time.
During Holden Half-Time, in case you missed any of the scintillating moments from the first 40 minutes, thankfully we were able to catch up on them courtesy of Australian Unions Half-Time Highlights.
And during the entire match, in case you missed anything at all you could see it again with the help of the Harvey Norman Replays.
But if you weren't paying attention, at game's end we were treated to the Tooheys New Highlights of the whole match, for the dummies whose memories were so short that they'd forgotten what happened in the past 80 minutes.
Impressively, one of the sponsors has even managed to have a new awarded added in its name.
Forget that Paul Gallen was the Player of the Series. Small fry. Corey Parker was the Australian Unions Hardest Working Player.
#OriginIII Tweets
I mean, I get the symbiosis - unions, hard working, league is apparently a working man's game - but are we suggesting the man of the match or the player of the series wasn't working their guts out too?
It almost felt as though the Hardest Working Player was the Origin equivalent of what we once used to call, in our schooluard cruelty, the "try-hard award" for the kid who was a bit unco-ordinated but gave it red-hot go anyway.
Even the commentators and Parker himself seemed to be embarrassed about presenting and accepting this award.
Still, if the TV coverage wasn't doing it for you, you were reminded that you could download Ninemsn's Jump-in app, which you could buy from the Apple App Store, and watch the game with the benefit of 16 different camera angles. Sixteen!
But in case you thought the app might allow you to escape the promotional guff being thrown at you on commercial TV, think again - a number of users complained online about the amount of ads they were bombarded with and had trouble skipping past.
Last year the big rugby league controversy - well, apart from the ubiquitous drunken off-field incidents - was the frequency with which Tom Waterhouse was popping up, and the way he had somehow become a guest commentator and not just a bloke running a betting company who was throwing a lot of money at Channel Nine.
Tom has disappeared from the commentary team this year, but if you wanted a sure bet for the evening, it was that his company had secured one of the first ads after the game was over.
All this brainwashing was starting to get to me a bit, but at least I could take my mind off it by planning what other marquee Channel Nine programs I would watch next - the network wasn't shy about pushing those either.
Now, I'm not naive. I know Origin (sorry, the Holden State of Origin) is big business and I'm not kidding myself that Channel Nine bought the TV rights to the tournament out of the goodness of its heart.
Sponsors want to be associated with the event and they will pay good money for the privilege of getting bang for their buck through as many mentions and pieces of exposure as possible.
And Nine is within its rights to use the opportunity to showcase its own programs (goodness knows Seven has assaulted us with enough "After the Tennis" or "After the Olympics" promos over the years).
I also understand that, from an advertiser's point of view, there's a huge captive audience watching almost three hours of football coverage and it's a great opportunity to get your message across.
And I understand that sport doesn't pay for itself - someone needs to pay the players and the league machine so we can all have the pleasure of seeing 26 blokes run around a paddock (although don't get me started on the huge gap between sporting salaries and other professions whose work actually achieves a lot more for society).
Really, I get it.
It's just that when you're on the other end of all of that advertising dollar and can't press the skip or fast-forward button, it takes a lot of the joy out of simply watching some sport.
Advertising isn't new in sport, least of all in football. We got used to sponsored jerseys long ago, we got used to the increased frequency of ads during big matches, we got used to everyone having to awkwardly thank the sponsors from the podium.
But at least if we're going to have to be subjected to all this, can it not be a bit more entertaining? Take a leaf out of the US Superbowl, whose ads are almost an event in themselves. They are clever, well planned, elaborate and often very funny.
Give me some of that as part of my Origin experience and I might not dread it so much.
Poll: Vote for your favourite State of Origin meme
Olympics not focus as rugby sevens series kicks off
editors picks opinion state of origin


After the 2018 international edition of Bar Convent Brooklyn in New York, BCB, one of the largest international meetings of the bar industry, takes the second step in its expansion strategy and will be held in Sao Paulo in 2019. BCB announced in a press release.
The event will take place on June 17 and 18, 2019 in São Paulo and will serve as a B2B (Business to Business) meeting platform for professionals from the bar and spirits industry and bar owners and barmen from the country. The Brazilian spirits industry has evolved in the last ten years and thus meets the demand of the country's nightlife.
The aim of the event is not only to transmit knowledge, trends, experiences and business perspectives and to develop relationships between industry professionals, but also to support the Brazilian bar industry in increasing its position on the list of the 100 best bars in the world. On the other hand, both international and local content will be shown to the bartending community across the country.