Non-Analyticity and the van der Waals Limit

We study the analyticity properties of the free energy fγ(m) of the Kac model at points of first order phase transition, in the van der Waals limit γ ↘ 0. We show that there exists an inverse temperature β0 and γ0 > 0 such that for all β ≥ β0 and for all γ ∈ (0, γ0), fγ(m) has no analytic continuation along the path m ↘ m∗ (m∗ denotes spontaneous magnetization). The proof consists in studying high order derivatives of the pressure pγ(h), which is related to the free energy fγ(m) by a Legendre transform.