AENC/resampling_chain Changeset - a43268439e66 · Centrum Wiskunde & Informatica (CWI)

Changeset - a43268439e66

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Andras Gilyen - 8 years ago 2017-09-11 19:55:55
gilyen@clayoquot.swat.cwi.nl

p_c characterization

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 &	0 & 1 & 2 & 3+2/3 & 6.44 & \cellcolor{blue!25}11.0 & 18.5 & 30.02 & 47.10 & 71.68 & 106.0 & 152.9 & 215.4 & 297.4 & 403.1 & 537.21 & 705.25 & 913.31 & 1168.2 & 1477.4 & 1849.1 \\
 &	0 & 1 & 2 & 3+2/3 & 6.44 & 11.0 & \cellcolor{blue!25}18.7 & 31.21 & 50.83 & 80.80 & 125.3 & 189.7 & 280.8 & 407.0 & 578.6 & 808.13 & 1110.2 & 1502.6 & 2005.6 & 2643.2 & 3443.1 \\
 &	0 & 1 & 2 & 3+2/3 & 6.44 & 11.0 & 18.7 & \cellcolor{blue!25}31.44 & 52.08 & 84.95 & 136.0 & 213.6 & 328.9 & 496.5 & 735.6 & 1070.7 & 1532.5 & 2159.5 & 2998.8 & 4108.1 & 5556.7 \\
 &	0 & 1 & 2 & 3+2/3 & 6.44 & 11.0 & 18.7 & 31.44 & \cellcolor{blue!25}52.30 & 86.27 & 140.7 & 226.3 & 358.4 & 558.4 & 855.4 & 1289.0 & 1911.5 & 2791.4 & 4017.2 & 5701.4 & 7985.9 \\
 &	0 & 1 & 2 & 3+2/3 & 6.44 & 11.0 & 18.7 & 31.44 & 52.30 & \cellcolor{blue!25}86.49 & 142.1 & 231.6 & 373.4 & 594.8 & 934.4 & 1447.1 & 2209.0 & 3324.6 & 4934.8 & 7226.9 & 10447. \\
             \vdots \\
-& 0 & 1 & 2 & 3+2/3 & 6.44 & 11.08 & 18.76 & 31.45 & 52.31 & 86.49 & 142.33 & 233.31 & 381.17 & 621.02 & \cellcolor{blue!25}1009.38 & 1637.13 & % 2650.74 & 4285.68 & 6913.55 & 11171.2 & 18052.2
+& 0 & 1 & 2 & 3+2/3 & 6.44 & 11.08 & 18.76 & 31.45 & 52.31 & 86.49 & 142.33 & 233.31 & 381.17 & 621.02 & 1009.38 & \cellcolor{blue!25}1637.13 & % 2650.74 & 4285.68 & 6913.55 & 11171.2 & 18052.2
         \end{tabular}
+	}
 	\end{table}
 	We observe that this is a power series in $p$. We discovered a very regular structure in this power series. It seems that for all $k\in\mathbb{N}$ and for all $n>k$ we have that $a^{(n)}_k$ is constant, this conjecture we verified using a computer up to $n=14$.
 	\newpage
 \end{comment}
+~
 Questions:
 \begin{itemize}
-	\item Can we generalise the proof to other translationally invariant spaces, like the torus?
+	\item Can we prove some upper bound of the coefficients in the difference, other than they are zero for small powers?
 	\item In view of this proof, can we better characterise $a_k^{(k+1)}$?
 	\item Why did Mario's and Tom's simulation show that for fixed $C$ the contribution coefficients have constant sign? Is it relevant for proving \ref{it:pos}-\ref{it:geq}?
 \end{itemize}
 	%I think the same arguments would translate to the torus and other translationally invariant spaces, so we could go higher dimensional as Mario suggested. Then I think one would need to replace $|S_{><}|$ by the minimal number $k$ such that there is a $C$ set for which $S\cup C$ is connected. I am not entirely sure how to generalise Lemma~\ref{lemma:probIndepNewGen} though, which has key importance in the present proof.
 \newpage
 \section{Characterisation of $p_c$}
 \textbf{Conjecture} for a fixed $p\in [0,1]$ the following are equivalent:
 \begin{enumerate}
 	\item $\lim_{n\to\infty}\P^{[-n,n]}_{\overline{\{0\}}}(\Z{\{n\}})>0$
 	\item $\P^{[-\infty,\infty]}_{\overline{\{0\}}}(\text{Not reaching the all 1 state})>0$
 	\item $\P^{[-\infty,\infty]}(\NZ{\{0\}})>0$
 	\item $\P^{[0,\infty]}(\NZ{\{0\}})>0$
 	\item $\lim_{n\to\infty}\P^{[0,n]}(\NZ{\{0\}})>0$
 	\item $\exists c,\lambda>0:\P^{[-\infty,\infty]}(\Z{[k]})<ce^{-\lambda k}$
 	\item $\exists c,\lambda>0:\mathrm{Cov}^{[-\infty,\infty]}(A,B)<ce^{-\lambda d(A,B)}$
 	\item $\exists c,\lambda>0\,\forall n\in\mathbb{N}:\mathrm{Cov}^{[n]}(A,B)<ce^{-\lambda d(A,B)}$
 	\item $R^{(\infty)}<\infty$
 \end{enumerate}
 \begin{proof}
 	$1\Leftrightarrow 2:$
 	\begin{align*}
 		\P^{[-\infty,\infty]}_{\overline{\{0\}}}(\text{Not reaching the all 1 state})>0
 		&=\P^{[-\infty,\infty]}_{\overline{\{0\}}}(\text{Resampling arbitrary far away})>0\\
 		&=\P^{[-\infty,\infty]}_{\overline{\{0\}}}\left(\bigcap_{n=1}^{\infty}\Z{\{-n\}}\cup\Z{\{n\}}\right)>0\\
 		&=\lim_{n\to\infty}\P^{[-\infty,\infty]}(\Z{\{-n\}}_{\overline{\{0\}}}\cup\Z{\{n\}})>0\\
 		&=\lim_{n\to\infty}\P^{[-n,n]}_{\overline{\{0\}}}(\Z{\{-n\}}\cup\Z{\{n\}})>0
 	\end{align*}
 \end{proof}
 \newpage
 \section{Quasiprobability method}
 Let us first introduce notation for paths of the Markov Chain
 \begin{definition}[Paths]
 	We define a \emph{path} of the Markov Chain as a sequence of states and resampling choices $\xi=((b_0,r_0),(b_1,r_1),...,(b_k,r_k)) \in (\{0,1\}^n\times[n])^k$ indicating that at time $t$ Markov Chain was in state $b_t\in\{0,1\}^n$ and then resampled site $r_t$. We denote by $|\xi|$ the length $k$ of such a path, i.e. the number of resamples that happened, and by $\mathbb{P}[\xi]$ the probability associated to this path.

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