OAI.RadialTransition.main

The theorem states that there exist a real number p, a pair potential φ : [0,∞) → ℝ, positive constants B and C, a function f : ℝ → ℝ, and a number βc such that the following all hold. First, p > 0 and p is the packing density: the maximal number of points in the cube [0,L]³ in ℝ³ with pairwise Euclidean distances at least 1, divided by L³, tends to p as L → ∞. Second, φ is bounded and continuous. Third, φ is stable with constant B, meaning that for every N and every configuration of N points in ℝ³ the energy, the sum of φ(|xᵢ−xⱼ|) over pairs i<j, is at least −BN. Fourth, φ has algebraic decay with constant C: |φ(r)| ≤ C r^(−3−1/32) for all r ≥ 1. Fifth, f is the canonical free energy of φ at density 5p/3: for every inverse temperature β > 0, with N = ⌊(5p/3)L³⌋ and partition function Z = (1/N!)∫ exp(−β·energy) over configurations of N points in the cube [0,L]³, the quantity −log Z/(βL³) tends to f(β) as L → ∞. Finally, βc lies in the closed interval [7/8, 9/8], and f has a strict corner at βc: f has a left derivative dLeft and a right derivative dRight at βc with dRight < dLeft, and f is not differentiable at βc. The theorem is stated with its proof admitted.