From 9b0f04598ca90324e643cc73cb0f7a8ee2570e33 2017-09-07 16:40:09
From: Andras Gilyen <gilyen@clayoquot.swat.cwi.nl>
Date: 2017-09-07 16:40:09
Subject: [PATCH] nicer proof

---

diff --git a/main.tex b/main.tex
index dc9ce03d46a66d3759fb20823dcd29614deadf4d..88768eec07d4f2bc9f68601c108984e59252153f 100644
--- a/main.tex
+++ b/main.tex
@@ -685,17 +685,17 @@ The intuition of the following lemma is that the far right can only affect the z
 			R^{(n)}
 			&= \E^{(n)}(\Res{1}) \tag{by translation invariance}\\
 			&= \sum_{k=1}^{\infty}\P^{(n)}(\Res{1}\geq 1) \\
-			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}\P^{(n)}(\Res{1}\geq 1\& P\in\mathcal{P}) \tag{partition}\\
-			&= \frac{1}{n}\sum_{v\in[n]}\sum_{t=1}^{\infty}\sum_{P\text{ patch}}t\cdot\P^{(n)}(v \text{ is resampled }t\text{ times and }v\in P | P\in\mathcal{P}) \; \P^{(n)}(P\in\mathcal{P})\\
-			&= \frac{1}{n}\sum_{P\text{ patch}}\E^{(n)}(\# \text{ resamples in }P|P\in \mathcal{P}) \; \P^{(n)}(P\in\mathcal{P})\\
-			&= \sum_{s=1}^{n-1}\E^{(n)}(\# \text{ resamples in }[s] \;|\; [s]\in \mathcal{P}) \; \P([s]\in\mathcal{P}) +\bigO{p^{n}}
-			\tag{by translation symmetry}\\
-			&= ???? \\
-			&= \sum_{s=1}^{n-1}\E^{[0,s+1]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})\P^{[s+1,n]}(\NZ{s+1}\cap\NZ{n})/(1+p)^2+\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\   
-			&= \sum_{s=1}^{n-1}\E^{[0,s+1]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})\left(\P^{[s+1,n]}(\NZ{s+1})\right)^2/(1+p)^2+\bigO{p^{n}} \tag{by Lemma~\ref{lemma:independenetSidesNew}}\\   
-			&= \sum_{s=1}^{n-1}\E^{[0,s+1]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})\left(\P^{[s+1,N]}(\NZ{s+1})\right)^2/(1+p)^2+\bigO{p^{n}} \tag{by Corollary~\ref{cor:probIndepNew}}\\   			
-			&= \sum_{s=1}^{n-1}\E^{[-N,N]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})+\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}, Corollary~\ref{cor:probIndepNew}}\\   	
-			&= \sum_{s=1}^{N}\E^{[-N,N]}(\# \text{ resamples in }[s]|[s]\in \mathcal{P})+\bigO{p^{n}}.
+			%&= \sum_{k=1}^{\infty}\sum_{\underset{\ell\geq r-1}{\ell,r\in[n]}}\P^{(n)}(\Res{1}\geq 1\,\&\, [\ell+1,r-1]\in\mathcal{P}) \tag{partition}\\
+			%&= \sum_{k=1}^{\infty}\sum_{\underset{\ell\geq r}{\ell,r\in[n]}}\P^{(n)}(\Res{1}\geq 1\,\&\, [\ell+1,r-1]\in\mathcal{P})  +\bigO{p^{n}} \\	
+			%&= \sum_{k=1}^{\infty}\sum_{\underset{\ell\geq r}{\ell,r\in[n]}}\P^{[l,r]}_{b_{\ell}=b_{r}=1}(\Res{1}\geq 1\,\&\, [\ell+1,r-1]\in\mathcal{P}) \P^{[r,\ell]}(\NZ{\ell,r}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\				
+			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}\P^{(n)}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) \tag{partition}\\
+			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{(n)}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) +\bigO{p^{n}}\\
+			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{[P\cup \partial P]}_{b_{\partial P}=1}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) \P^{[\overline{P}]}(\NZ{\partial P}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\
+			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{[P\cup \partial P]}_{b_{\partial P}=1}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) \left(\left(\P^{[|\overline{P}|]}(\NZ{1})\right)^2+\bigO{p^{|\overline{P}|}}\right) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:independenetSidesNew}}\\
+			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{[P\cup \partial P]}_{b_{\partial P}=1}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) \left(\left(\P^{[N]}(\NZ{1})\right)^2+\bigO{p^{|\overline{P}|}}\right) +\bigO{p^{n}} \tag{by Corollary~\ref{cor:probIndepNew}}\\
+			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}^{|P|<n}\P^{[-N,N]}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\
+			&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}\P^{[-N,N]}(\Res{1}\geq 1\,\&\, P\in\mathcal{P}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\
+			&= \E^{[-N,N]}(\Res{1})+\bigO{p^{n}}.
 		\end{align*}		
 		
 		Repeating the same calculation with $m$, and comparing the two expressions completes the proof.