The central nilradical of nonnoetherian dimer algebras

Let $Z$ be the center of a nonnoetherian dimer algebra $A$ on a torus. We show that the nilradical $\operatorname{nil}Z$ of $Z$ is prime, may be nonzero, and consists precisely of the central elements that vanish under a cyclic contraction of $A$. This implies that the nonnoetherian scheme $\operatorname{Spec}Z$ is irreducible. We also show that the reduced center $\hat{Z} = Z/\operatorname{nil}Z$ embeds into the center $R$ of the corresponding ghor algebra, and that their normalizations are equal. Finally, we give three characterizations of the normality of $R$, and show that if $\hat{Z}$ is normal, then it has the special form $k + J$ where $J$ is an ideal of the cycle algebra of $A$.