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We perform large scale simulations to characterize the transition in quenchedQCD. It is shown by a rigorous finite size scaling that the transition is offirst order. After this qualitative feature quantitative results are obtainedwith unprecedented precision: we calculate the transition temperature$w_0T_c$=0.25384(23), -- which is the first per-mill accurate result in QCDthermodynamics -- and the latent heat $\Delta E/T_c^4$=1.025(21)(27) in bothcases carrying out controlled continuum and infinite volume extrapolations. Asit is well known the cost of lattice simulations explodes in the vicinity ofphase transitions, a phenomenon called critical slowing down for second orderphase transitions and supercritical slowing down for first order phasetransitions. We show that a generalization of the parallel tempering algorithmof Marinari and Parisi [Europhys. Lett. 19, 451 (1992)] originally for spinsystems can efficiently overcome these difficulties even if the transition isof first order, like in the case of QCD without quarks, or with very heavyquarks. We also report on our investigations on the autocorrelation times andother details.

Precision study of the continuum SU(3) Yang-Mills theory: how to use  parallel tempering to improve on supercritical slowing down for first order  phase transitions