= Solution
For finite $p=4$, the <clock model> has a discrete symmetry. Domain walls have finite energy per unit boundary area, so thermal disorder destroys long-range order in one dimension but a finite-temperature ordered phase can exist in two dimensions. Its <lower critical dimension> is therefore $\boxed{d_{\mathrm l}=1}$.
As $p\to\infty$, the permitted angles become continuous and the model becomes the <XY model> with $O(2)$ symmetry. The <Mermin-Wagner theorem> forbids spontaneous long-range order at positive temperature in two dimensions, so the lower critical dimension for conventional symmetry breaking is $\boxed{d_{\mathrm l}=2}$. The two-dimensional model can nevertheless undergo a <Berezinskii–Kosterlitz–Thouless transition> between algebraic and exponential correlation decay.
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