Gap Statistics for Confined Particles with Power-Law Interactions

Dynamical fluctuations in the Riesz gas

We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the |x-y|^{-s} Riesz potential, causing the overdamped motion of particles. We investigate fluctuations of the integrated current and the position of a tagged particle. We show that for 0<s<1, the standard deviations of both quantities grow as t^{s/2(1+s)}. When s>1, the interactions are effectively short-ranged, and the universal subdiffusive t^{1/4} growth emerges with only amplitude depending on the exponent s. We also show that the two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion.

Finite-temperature equilibrium density profiles of integrable systems in confining potentials

We study the equilibrium density profile of particles in two one-dimensional classical integrable models, namely hard rods and the hyperbolic Calogero model, placed in confining potentials. For both of these models the interparticle repulsion is strong enough to prevent particle trajectories from intersecting. We use field theoretic techniques to compute the density profile and their scaling with system size and temperature, and we compare them with results from Monte Carlo simulations. In both cases we find good agreement between the field theory and simulations. We also consider the case of the Toda model in which interparticle repulsion is weak and particle trajectories can cross. In this case, we find that a field theoretic description is ill-suited and instead, in certain parameter regimes, we present an approximate Hessian theory to understand the density profile. Our work provides an analytical approach toward understanding the equilibrium properties for interacting integrable systems in confining traps.