For the unit-weight cycle \(C_n\), \(n\ge 3\), let \(c>0\) and define \(\omega >0\) by \(2\cosh \omega =2+c\). Then \[\det (cI+L_{C_n})=2\cosh (n\omega )-2 =4\sinh ^2(n\omega /2).\] This uses the nearest-neighbor Laplacian \(L_{C_n}=2I-T_n-T_n^{-1}\), with every undirected edge counted once. The condition \(n\ge 3\) avoids replacing the two-vertex cycle by a different parallel-edge convention.

Now fix \(\mu >0\), set \(c=\mu /n^2\), and write \(\omega _n\) for the corresponding solution. Since \(2\cosh x-2=x^2+O(x^4)\) at zero, \(\omega _n\to 0\) and \(n\omega _n\to \sqrt \mu \). Therefore \[\det (\mu n^{-2}I+L_{C_n}) \longrightarrow 4\sinh ^2(\sqrt {\mu }/2).\] For the circle mesh, with \(L_n=n^2L_{C_n}\) and \(Q_n=L_n+\mu I\), this is exactly \[\frac {\det Q_n}{n^{2n}} \longrightarrow 4\sinh ^2(\sqrt {\mu }/2),\] or, in the induced-action notation, \[\frac 12\log \det Q_n-n\log n \longrightarrow \log \left (2\sinh (\sqrt {\mu }/2)\right ).\] The determinant here is that of the operator \(Q_n\), equivalently its matrix in an orthonormal basis for the supplied normalized pairing. In raw vertex coordinates the action-form matrix is \(Q_n/n\); its determinant carries the additional factor \(n^{-n}\). Thus a reference measure convention still matters when passing from this operator calculation to a Gaussian integral.

This finite-product limit complements the fixed-mode covariance limit and the exact cycle covariance. It does not define an infinite product without a normalization, prove convergence of full-field measures, or identify the limit with a continuum determinant under an unstated regularization convention. The positive mass is fixed throughout.