Simon's Problem 7 (1984): a continuum phase transition for a stable tempered pair potential, on an open interval of densities

Algebraic-decay manuscript, Theorem 1.1: there is a bounded continuous radial $\phi$ on $\mathbb R^3$ with $\sum_{i<j}\phi\ge-BN$ and $|\phi(r)|\le Cr^{-3-1/32}$ ($r\ge1$), an open interval $I$ centred at $5p/3$ ($p$ the unit packing density) and one $\beta_c\in[7/8,9/8]$ such that for every $\rho\in I$ the canonical free energy exists for all $\beta>0$ and $f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho)$. The companion gives a potential with divergent core and $o(r^{-3})$ tail, a common $\beta_c\in(1/2,3/2)$ on an open density interval, and says it does not meet a fixed power margin. Not shown: a transition for every sufficiently large $\rho$ as Simon's wording asks, for Lennard-Jones or any finite-range potential, or identification of the phases. Earlier continuum transitions (Widom-Rowlinson, Lebowitz-Mazel-Presutti, recent Kac-type models) used several species, many-body terms or box-dependent interactions.

Algebraic-decay manuscript, Theorem 1.1: there is a bounded continuous radial $\phi$ on $\mathbb R^3$ with $\sum_{i<j}\phi\ge-BN$ and $|\phi(r)|\le Cr^{-3-1/32}$ ($r\ge1$), an open interval $I$ centred at $5p/3$ ($p$ the unit packing density) and one $\beta_c\in[7/8,9/8]$ such that for every $\rho\in I$ the canonical free energy exists for all $\beta>0$ and $f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho)$. The companion gives a potential with divergent core and $o(r^{-3})$ tail, a common $\beta_c\in(1/2,3/2)$ on an open density interval, and says it does not meet a fixed power margin. Not shown: a transition for every sufficiently large $\rho$ as Simon's wording asks, for Lennard-Jones or any finite-range potential, or identification of the phases. Earlier continuum transitions (Widom-Rowlinson, Lebowitz-Mazel-Presutti, recent Kac-type models) used several species, many-body terms or box-dependent interactions.