I am reading a [paper](https://link.springer.com/content/pdf/10.1007/s11222-015-9598-x.pdf) about constructing a non-reversible Metropolis Hastings Markov chain from a reversible one as described at a high level in paragraph $3$ of page $1$.



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> But I don't understand how, given a reversible Markov chain $P$ with stationary distribution $\pi$, a non-reversible Markov chain is constructed with the same stationary distribution. This is explained in Section $2.3$.
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It says the idea is, given a non-reversible Markov chain $P$ with stationary distribution $\pi$, to consider the vorticity matrix $\Gamma$ of $P$ defined (in equation $(1)$) as:


$$\Gamma(x,y):=\pi(x)P(x,y)-\pi(y)P(y,x)$$


In Section $2.3$, it says consider a Markov chain $Q$ and a vorticity matrix $\Gamma$, then it defines the non-reversible Hastings ratio as


$$R\_{\Gamma}(x,y):=\begin{cases} 
\frac{\Gamma(x,y)+\pi(y)Q(y,x)}{\pi(x)Q(x,y)}\quad\text{if}\quad \pi(x)Q(x,y)\neq 0 \\
1 \quad\text{otherwise}
\end{cases}
$$


and the acceptance probabilities as $A\_\Gamma(x,y) = \min\left(1,R\_\Gamma(x,y) \right)$
and finally the transition probabilities $P\_{\Gamma}$ of the non-reversible Metropolis Hastings in $(9)$. Let me breakdown my misunderstanding in two questions.



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> $(1)$ Firstly, is $\Gamma$ the vorticity matrix of $Q$ as it is defined in Section $2.3$?
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I think it is because the inequality under $(8)$ says $\Gamma(x,y)\leq \pi(x)Q(x,y)$ which would be true only if $\Gamma$ is the vorticity matrix of $Q$ because if $\Gamma(x,y):=\pi(x)P(x,y)-\pi(y)P(y,x)$, then $\Gamma(x,y)\leq \pi(x)Q(x,y)$.



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> $(2)$ If the answer to $(1)$ is yes, then how are we constructing a non-reversible Markov chain from a reversible one? Because $\Gamma\neq 0$ iff $Q$ is non-reversible and thus we are constructing a non-reversible chain $P\_{\Gamma}$ from a non-reversible chain $Q$. Isn't that correct?
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About non-reversible Metropolis Hastings Markov chain
As defined after lemma 2.1, any skew-symmetric matrix $\Gamma$ (i.e. $\Gamma=-\Gamma^T$) that satisfies $\Gamma {\mathbb 1}=0$ is called a vorticity matrix. It need not be of the form $\pi(x)P(xy)-\pi(y)P(y,x)$.


