Abstract
We study the pressure, $P$, of $\mathrm{SU}(N)$ gauge theory on a two-dimensional torus as a function of area, $A=l/t$. We find a crossover scale that separates the system on a large circle from a system on a small circle at any finite temperature. The crossover scale approaches zero with increasing $N$ and the crossover becomes a first-order transition as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and $l\ensuremath{\rightarrow}0$ with the limiting value of $\frac{2Pl}{(N\ensuremath{-}1)t}$ depending on the fixed value of $Nl$.
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