This paper is a sequel to [author, ibid. 27, No. 3, 291-383 (1991;

Zbl 0746.60024)]. It has been drawn from the author’s thesis at the University Paris-Sud Orsay, March 1990, supervised by R. Azencott. Aim of this second part is to study applications of large deviations to cooling systems of the critical type, i.e. $1/T\sb n=(1/d) \ln n+B+o(1)$, where $d$ is Hajek’s critical depth. Although quasi-equilibrium is not maintained for such schedules, it turns out that the law of the system is not “too far” from quasi-equilibrium if $B$ is small. However, if $B$ is above some critical value, convergence rates of the annealing algorithm can be made arbitrarily poor by increasing $B$. Sharp large deviations estimates are needed in order to obtain the desired results. - -- Contents: 1. Estimation of the probability of the critical cycle. 2. Asymptotics of the law of the system. 3. Triangular cooling schedules. 4. The optimization problem far from the horizon.