Uniform displacement bounds and Gibbs limits for periodic one-dimensional Riesz gases

For the neutral periodic one-dimensional Riesz gas with pair potential locally $-|x|^a$, $0<a<1$, we prove a particle-displacement variance bound of order $β^{-1}$, uniformly in the number of particles. Log-concavity also gives exponential displacement tails. Every stationary periodic thermodynamic