\section{Subaditivity tricks}

\let\lam\lambda
This chapter concerns itself mainly with one theorem that uses
a subaditivity property.

In a subcritical setting ($0 < \lam < \lam_c$), we define a 
function $\xi(\lam) \equiv \lim_{n \Rightarrow \infty} (-n^{-1} log p_n(\lam))$.
Equivalently, this can be defined as
$\xi(\lam) = \frac{-1}{n} \sum_{k=n}^{\infty} p_k(\lam)$.

\begin{thm}
Function $\xi$, as defined in the previous paragraph, is a proper function
and moreover it continously reaches infinity as $\lam$ goes to zero.
\end{thm}

\begin{proof}

First, we will investigate the definition of the functions $p_k(\lam)$.
These can be equivalently expressed as

$$ p_{k+1}(\lam) = (k+1) \lam^k \int \int \cdots \int h(0, \vec{x})
\ \exp(-\lam A(0, \vec{x}))\ d(\vec{x}),$$

where the two new functions are the ``indicator function'' $h$ and an ``area
function'' $A$. Notice the connection between $p_k$ and the Poisson distribution.

More formally, using the Iverson notation:
$$h(0, x) \equiv [G(\vec{x};1) {\rm\ is\ connected}],$$
$$A(0, x) \equiv \bigcup_{i=0}^k {\rm Ball}(x_i, 1).$$

If we multiply a $p_{k+1}(\lam)$ with another $p_{j+1}(\lambda)$, we can notice
that the indicator functions merge easily, since the two sets of points have a common
element, the origin:
$$h(0,\vec{x}) h(0,\vec{y}) = h(0,\vec{x},\vec{y}).$$

As for the merge functions, we cannot achieve equality here, but since they
calculate volume of an area, we have:
$$ A(\vec{x}) + A(\vec{y}) \ge A(\vec{x}, \vec{y}). $$

Substituting the two bounds, we get:
$$ p_{k+1}(\lam) + p_{j+1}(\lam) \le (k+1)(j+1) \lam^{k+j} \int \int \cdots \int
h(0,\vec{x}, \vec{y})\ \exp(-\lam A(\vec{x}, \vec{y}))\ d\vec{x} d\vec{y}.$$

If we define $q_k \equiv - \log p_k$, we get a nice symbolic notation:
$$ q_{m+n} \le q_m + q_n. $$

\end{proof}