Low-temperature crystallization conjecture for one-dimensional Riesz gases

Let $\mathsf{P}_{s,\beta}$ denote a one-dimensional non-singular Riesz gas with homogeneity parameter $s$, and let $\mathsf{Leb}_{\mathbb{R}}$ be Lebesgue measure on $\mathbb{R}$. For point processes or measures, write $\mathsf{Wass}_1$ for the Wasserstein-1 distance. Riesz-gas crystallization conjecture. For $-1<s<0$, there exists $\beta_0(s)>0$, decreasing in $s$, such that for $\beta>\beta_0(s)$ we have

$$\mathsf{Wass}_1(\mathsf{P}_{s,\beta},\mathsf{Leb}_{\mathbb{R}})<+\infty,$$

whereas for $0<\beta<\beta_0(s)$ we have

$$\mathsf{Wass}_1(\mathsf{P}_{s,\beta},\mathsf{Leb}_{\mathbb{R}})=+\infty.$$

The conjecture predicts a temperature-driven transition between a crystalline low-temperature regime and a quasi-solid high-temperature regime for one-dimensional Riesz gases with $-1<s<0$. It would imply the three rigidity properties discussed in the paper, with the Cyclic-Factor property corresponding to crystallization; the phase transition has not yet been proved rigorously.