Failure of Ornstein-Zernike asymptotics for the pair correlation function at high temperature and small density

We report on recent results that show that the pair correlation function of systems with exponentially decaying interactions can fail to exhibit Ornstein-Zernike asymptotics at all sufficiently high temperatures and all sufficiently small densities. This turns out to be related to a lack of analyticity of the correlation length as a function of temperature and/or density and even occurs for one-dimensional systems.

Figure 1

Local parametrization of $\partial \mathcal{U}$.

Figure 2

Sketches of a typical path $\gamma$ in both regimes. Left: $\gamma$ consists of one giant step for $\beta <{\beta }_{\mathrm{sat}}\left(\widehat{x}\right)$. Right: $\gamma$ consists of $\mathsf{O}\left(‖x‖\right)$ small steps for $\beta >{\beta }_{\mathrm{sat}}\left(\widehat{x}\right)$.