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Zeros of the Potts Model Partition Function in the Large-$q$ Limit

We study the zeros of the $q$-state Potts model partition function $Z(Λ,q,v)$ for large $q$, where $v$ is the temperature variable and $Λ$ is a section of a regular $d$-dimensional lattice with coordination number $κ_Λ$ and various boundary conditions. We consider the simultaneous thermodynamic limit and $q \to \infty$ limit and show that when these limits are taken appropriately, the zeros lie on the unit circle $|x_Λ|=1$ in the complex $x_Λ$ plane, where $x_Λ=v q^{-2/κ_Λ}$. For large finite sections of some lattices we also determine the circular loci near which the zeros lie for large $q$.