The authors develop a probabilistic approach to the study of the equilibrium measure of systems with exclusions – such as hard-core gases, contours, polymers or animals – in the law-density or extreme-temperature regime. This regime has traditionally been studied via cluster-expansion methods, which relied either on sophisticated combinatorial estimations [V. A. Malyshev, Russ. Math. Surv. 35, 1-62 (1980); E. Seiler, “Gauge theories as a problem of constructive quantum field theory and statistical mechanics” (Berlin, 1982); D. C. Brydges, in: Critical phenomena, random systems, gauge theories, Pt. 1, 129-183 (1986; Zbl 0659.60136)] or an asute inductive hypotheses plus complex analysis [B. Kotecký and D. Preiss, Commun. Math. Phys. 103, 491-498 (1986; Zbl 0593.05006); R. Dobrushin, in: Topics in statistical and theoretical physics. Transl., Ser. 2, Am. Math. Soc. 177(32), 59-81 (1996; Zbl 0873.60074)]. The authors realize the equilibrium measure as the invariant measure of a loss network process whose existence is ensured by a subcriticality condition of a dominant branching process. In this regime the approach yields, besides existence and uniqueness of the measure, properties such as exponential space convergence and mixing, and a central limit theorem. The loss network converges exponentially fast to the equilibrium measure without metastable traps. This convergence is faster at low temperature, where it leads to the proof of an asymptotic Poisson distribution of contours.