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We consider the $N$ particle classical Riesz gas confined in aone-dimensional external harmonic potential with power law interaction of theform $1/r^k$ where $r$ is the separation between particles. As special limitsit contains several systems such as Dyson's log-gas ($k\to 0^+$),Calogero-Moser model ($k=2$), 1d one component plasma ($k=-1$) and the hard-rodgas ($k\to \infty$). Despite its growing importance, only large-$N$ fieldtheory 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 gapbetween successive particles. This quantity is analogous to the well-knownlevel spacing statistics which is ubiquitous in several branches of physics. Weshow that the variance goes as $N^{-b_k}$ and we find the $k$ dependence of$b_k$ via direct Monte Carlo simulations. We provide supporting arguments basedon microscopic Hessian calculation and a quadratic field theory approach. Wecompute the gap distribution and study its system size scaling. Except in therange $-1<k<0$, we find scaling for all $k>-2$ with both Gaussian andnon-Gaussian scaling forms.

Gap Statistics for Confined Particles with Power-Law Interactions