Gap Statistics for Confined Particles with Power-Law Interactions

We consider the N particle classical Riesz gas confined in a one-dimensional external harmonic potential with power-law interaction of the form 1/r(k), where r is the separation between particles. As special limits it contains several systems such as Dyson's log-gas (k -> 0(+)), the Calogero-Moser model (k = 2), the 1D one-component plasma (k = -1), and the hard-rod gas (k -> infinity). Despite its growing importance, only large-N field theory and average density profile are known for general k. In this Letter, we study the fluctuations in the system by looking at the statistics of the gap between successive particles. This quantity is analogous to the well-known level-spacing statistics which is ubiquitous in several branches of physics. We show that the variance goes as N-bk and we find the k dependence of b(k) via direct Monte Carlo simulations. We provide supporting arguments based on microscopic Hessian calculation and a quadratic field theory approach. We compute the gap distribution and study its system size scaling. Except in the range -1 < k < 0, we find scaling for all k > -2 with both Gaussian and non-Gaussian scaling forms.