between the identical universality class?
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This section considers a $d$-dimensional stochastic system of hard core particles with dynamics summarized in Table \[Table:DP\]. In Ref. [@PP05c], it is shown that two different stochastic systems modeled by tilded and untilded rates are equivalent if transition rates satisfy the relations $$\begin{aligned}
\tilde \sigma = \mu \sigma&,\qquad 2  \tilde D + \tilde \xi = 2  D + \xi,\\
\tilde D + \frac{\tilde \sigma}{2} &= D + \frac{\sigma}{2},\\
\tilde \lambda + \frac{\tilde \eta}{2} + \frac{\tilde \sigma}{2} &=
\frac{1}{\mu}\left (\lambda + \frac{\eta}{2}+\frac{\sigma}{2}\right ),\\
\tilde \lambda + \tilde \eta + \tilde \sigma - 2 \tilde D  &= \lambda +
\eta+\sigma - 2 D ,
\end{aligned}
\label{Eq:prime_unprime}$$ where any positive number $\mu$ which renders all tilded and untilded transition rates be nonnegative is physically meaningful. By equivalence is meant that all correlation functions of the model with tilded parameters can be deduced from those of the model with untilded parameters and vice versa. For instance, the particle density $\rho$ at time $t$ of two different stochastic many body systems becomes [@PP05c] $$\rho(D,\lambda,\eta,\xi,\sigma;\rho_0,t) = \frac{1}{\mu} \tilde \rho(\tilde D,
\tilde \lambda,\tilde \eta,\tilde \xi,\tilde \sigma;\tilde \rho_0,t),
\label{Eq:rho_relation}$$ if the initial density of two systems has the relation $\tilde
\rho_0 = 