\begin{align*}

        \P_{I}(\Z{j})       &= \mathcal{O}(p^{d(i_*,j)}) \\

        \P_{I}(\Z{j},\NZ{s}) &= \mathcal{O}(p^{\dist{s}(i_*,j)})

    \end{align*}

    simply because a chain of zeroes has to be formed between $i_*$ and $j$, and in the second case this chain can not go through $s$. Now assume both statements hold up to $|I|-1$, then we prove them both for sets of size $|I|$.

    When we refer to an interval $[a,b]$ on the circle we could be referring to two possible intervals because of the periodicity of the circle. Define $[a,b]_j$ as the interval with vertex $j$ on the \emph{inside}. Furthermore by $-a$ we mean the vertex $n-a$, as one would expect modulo $n$.

    %If we refer to only $[a,b]$ then we mean $\{a,a+1,...,b\}$ where numbers are considered modulo $n$. So $[a,b]$ and $[b,a]$ are different intervals that cover the circle together.

    Without loss of generality, we can assume that $0=j < i_* < s < n$. We will now consider intervals around vertex 0.

    For $l,r\geq 1$ and $l+r\leq n$, define the event ``zeroes patch'' $\mathrm{ZP}^{[-l,r]_0}$ as the event of getting zeroes inside the interval $[-l,r]_0$ but not on the boundary, i.e.

    $$\mathrm{ZP}^{[-l,r]_0} = \NZ{-l} \cap \Z{-l+1} \cap \cdots \cap \Z{0} \cap \cdots \cap \Z{r-1} \cap \NZ{r}$$

    Claim:

    \begin{align*}

        \P_{I}(\mathrm{ZP}^{[-l,r]_0}) &= \P_{I'}(\mathrm{ZP}^{[-l,r]_0})

    \end{align*}

    The following diagram illustrates the situation

    \begin{center}

        \includegraphics{diagram_circle_lemma.pdf}

    \end{center}

    Note that by Claim~\ref{claim:eventindependence} we have

    \begin{align*}

        \P_{I}(\mathrm{ZP}^{[-l,r]_0}) = \P_{I \cap [-l,r]_0}(\mathrm{ZP}^{[-l,r]_0}) \;\cdot\; \P_{I\setminus [-l,r]_0}(\NZ{a},\NZ{b})

    \end{align*}

    We have $i_*\in I \setminus[-l,r]_0$, and $I\cap[-l,r]_0 = I' \cap [-l,r]_0$. Define $J=I\setminus[-l,r]_0$ and $J'=I'\setminus[-l,r]_0$. We have $|J|<|I|$ so we can apply the induction hypothesis to $J$:

    \begin{align*}

        \P_{J}(\NZ{-l},\NZ{r})

        &=

        + \mathcal{O}(p^{\min\left( \dist{r}(i_*,-l) , d(i_*,r)\right)+1-|J|})

    \end{align*}

    Note that the event $\mathrm{ZP}^{[-l,r]_0}$ contains $l+r-1$ zeroes, so $\P_{I \cap [-l,r]_0}(\mathrm{ZP}^{[-l,r]_0}) = \mathcal{O}(p^{l+r-1-|I\cap[-l,r]_0|})$. This means

    \begin{align*}

        \P_{I}(\mathrm{ZP}^{[-l,r]_0})

        &= \P_{I' \cap [-l,r]_0}(\mathrm{ZP}^{[-l,r]_0})

        \left( \P_{I' \setminus [-l,r]_0}(\NZ{a},\NZ{b}) + \mathcal{O}(p^{\min\left( \dist{r}(i_*,-l) , d(i_*,r)\right)+1-|J|}) \right) \\

        &= \P_{I' \cap [-l,r]_0}(\mathrm{ZP}^{[-l,r]_0}) \;\cdot\; \P_{I'\setminus [-l,r]_0}(\NZ{a},\NZ{b}) \\

        &\qquad + \mathcal{O}(p^{\min\left( \dist{r}(i_*,-l) , d(i_*,r)\right)+1-|J| + l+r-1-|I\cap[-l,r]_0|}) \\

        &= \P_{I'}(\mathrm{ZP}^{[-l,r]_0})

        + \mathcal{O}(p^{\min\left( \dist{r}(i_*,-l) , d(i_*,r)\right)+l+r-|I|})

    \end{align*}

    \begin{align*}

        \P_{I}(\Z{0})

        &=\sum_{\substack{l,r\geq 1\\l+r\leq n}}

        \P_I(\mathrm{ZP}^{[-l,r]_0})

        \tag{the events are a partition of $\Z{0}$}\\

        &=\sum_{\substack{l,r\geq 1\\l+r\leq n}}

        \P_{I'}(\mathrm{ZP}^{[-l,r]_0})

        &= \P_{I'}(\Z{0}) + \mathcal{O}(p^{d(i_*,0)+1-|I|})

    \end{align*}

    Similarly, we have

    \begin{align*}

        \P_{I}(\Z{0} , \NZ{s})

        &=\sum_{l=1}^{n-s}\sum_{r=1}^{s}

        \P_{I}(\mathrm{ZP}^{[-l,r]_0},\NZ{s})

        \tag{partition of $\Z{0}$}\\

        &=\sum_{l=1}^{n-s}\sum_{r=1}^{s}

        &= \P_{I'}(\Z{0} , \NZ{s})

    \end{align*}

\end{proof}

\begin{definition}[Connected patches]

	Let $\mathcal{P}\subset 2^{\mathbb{Z}}$ be a finite system of finite subsets of $\mathbb{Z}$. We say that the patch set of a resample sequence is $\mathcal{P}$,

	if the connected components of the vertices that have ever become $0$ are exactly the elements of $\mathcal{P}$. We denote by $A^{(\mathcal{P})}$ the event that the set of patches is $\mathcal{P}$. For a patch $P$ let $A^{(P)}=\bigcup_{\mathcal{P}:P\in \mathcal{P}}A^{(\mathcal{P})}$.

\end{definition} 

Note by Tom: So $A^{(\mathcal{P})}$ is the event that the set of all patches is \emph{exactly} $\mathcal{P}$ whereas $A^{(P)}$ is the event that one of the patches is equal to $P$ but there can be other patches as well.

\begin{definition}[Conditional expectations]

	Let $S\subset\mathbb{Z}$ be a finite slot configuration, and for $f\in\{0,1'\}^{|S|}$ let $I:=S(f)$ be the set of vertices filled with particles. 

	Then we define

	$$R_I:=\mathbb{E}[\#\{\text{resamplings when started from inital state }I\}].$$