The high temperature phase transition in SU(N) gauge theories

We calculate the continuum value of the deconfining temperature in units of the string tension for SU(4), SU(6) and SU(8) gauge theories, and we recalculate its value for SU(2) and SU(3). We find that the $N$-dependence for $2 \leq N \leq 8$ is well fitted by $T_c/\sqrt{sigma} = 0.596(4) + 0.453(30)/N^2$, showing a rapid convergence to the large-N limit. We confirm our earlier result that the phase transition is first order for $N \geq 3$ and that it becomes stronger with increasing $N$. We also confirm that as $N$ increases the finite volume corrections become rapidly smaller and the phase transition becomes visible on ever smaller volumes. We interpret the latter as being due to the fact that the tension of the domain wall that separates the confining and deconfining phases increases rapidly with $N$. We speculate on the connection to Eguchi-Kawai reduction and to the idea of a Master Field.