  In the Sachdev-Ye-Kitaev model with generic order $q \ge 4$ random couplings,
we compute the critical temperature relating the Majorana fermions high
temperature perturbative vacuum to the vacuum where the replica diagonal
collective field $G(\tau, \tau')$ condenses. We study, by a finite temperature
diagrammatic analysis, the effective action of an auxiliary
Hubbard-Stratonovich bilocal field related to $G(\tau, \tau')$ in the large $N$
limit. Subtelties that arise in switching from the operatorial to the
functional integral representation of the SYK thermal partition function are
also discussed.
