Metropolis–Hastings Monte Carlo near the critical point T_c = 2/ln(1+√2)
The 2D Ising model (Onsager 1944) has exact critical temperature T_c = 2/ln(1+√2) ≈ 2.269. At T_c: correlation length diverges, magnetization ~ |T−T_c|^β with β=1/8, susceptibility ~ |T−T_c|^{-7/4}. Metropolis algorithm samples the Boltzmann distribution P(σ) ∝ e^{−H/T} ergodically.