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    \textbf{New conjecture(s)}: The following statements are equivalent

    \begin{enumerate}

        \item $p<p_c$

        \item (requires continuous time) $\P^{[0,\infty]}(\NZ{0}) > 0$

        \item (requires continuous time) $\P^{[-\infty,\infty]}(\NZ{0}) > 0$

        \item (requires continuous time) $\exists c,\lambda$ such that $\P^{[-\infty,\infty]}(\Z{[0,k]}) \leq c \; e^{-\lambda \; k}$

        \item (requires continuous time) $\exists c,\lambda$ such that $\mathrm{COV}^{[-\infty,\infty]}(A,B) \leq c \; e^{-\lambda \; d(A,B)}$

        \item (requires continuous time) $\E^{(\infty)}(\text{\# resamples of }0) < \infty$

        \item (first lines requires continuous time)

            \begin{align*}

                \P^{[-\infty,\infty]}(\textsc{NonStop})

                &= \P^{[-\infty,\infty]}(\textsc{NonStop} \text{ and unbounded}) \\

                &= \P^{[-\infty,\infty]}(\bigcap_{n=1}^{\infty} \Z{[-n,n]}) \\

                &= \lim_{n\to\infty} \P^{[-\infty,\infty]}(\Z{[-n,n]}) \\

                &\overset{?}{=} \lim_{n\to\infty} \P^{[-n,n]}(\Z{[-n,n]})

            \end{align*}

        \item (does \emph{not} requires continuous time) $\P^{[-\infty,\infty]}(\textsc{NonStop} \mid \text{start with single 0 at 0})$

        \item $\lim_{n\to\infty} \P^{[0,n]}(\Z{n} \mid \text{start with single 0 at 0}) = 0$

        \item $\lim_{n\to\infty} \P^{[0,n]}(\NZ{0}) > 0$ (note: symbolically computable for small n)

        \item $\lim_{n\to\infty} \P^{[0,n]}(\Z{0}) < 1$ (note: symbolically computable for small n)

        \item For the non-terminating process (resample a random 1 in case there are only 1s available), the number of zeroes in the stationary distribution is $o(n)$.

    \end{enumerate}

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