I want to find the $4$x$4$ Probability Transition Matrix under the temperature parameter T=2 of Metropolis Hastings. I know that, if x and y are neighbors,
$p(x,y) =$ $$ f(x) = \left\{ \begin{array}{ll} \frac{1}{N} & \quad \Delta E(x,y) \leq 0 \\ \frac{1}{N}*e^{-\Delta \frac{E(x,y)}{T}} & \quad \Delta E(x,y) > 0 \end{array} \right. $$
but I'm not sure what N represents.
If I want, say, $p(1,2)$, I know that the probability is $\frac{1}{N}$ because the change in energy is negative, but I'm not sure what $N$ is and my textbook does not specify.
$N$ is the number of neighbours. You select one of the $N$ neighbours uniformly, with equal probability $\frac1N$, and then you move to that neighbour with probability $1$ if it has lower energy, and with probability given by the exponential factor if it has higher energy.