Abstract
The two-dimensional one-component plasma is an ubiquitous model for several vortex systems. For special values of the coupling constant $\beta q^2$ (where $q$ is the particles charge and $\beta$ the inverse temperature), the model also corresponds to the eigenvalues distribution of normal matrix models. Several features of the system are discussed in the limit of large number $N$ of particles for generic values of the coupling constant. We show that the statistics of a class of radial observables produces a rich phase diagram, and their asymptotic behaviour in terms of large deviation functions is calculated explicitly, including next-to-leading terms up to order 1/N. We demonstrate a split-off phenomenon associated to atypical fluctuations of the edge density profile. We also show explicitly that a failure of the fluid phase assumption of the plasma can break a genuine $1/N$-expansion of the free energy. Our findings are corroborated by numerical comparisons with exact finite-N formulae valid for $\beta q^2=2$.
Highlights
Introduction and Main ResultsIn recent years there has been a considerable interest in the study of systems with logarithmic interactions
Our findings can be summarized as follows: 1. We show that the probability density function PN(∞)(x) =
In the derivation of our results, it will be clear that: a) The fluctuations for x ≤ 1 are driven by a collective behaviour of the charges; on the contrary, the fluctuations to the right x > 1 are associated to a spontaneous symmetry breaking in the problem. This change of behaviour of the Coulomb gas is at the heart of the change of speed in the large deviations tails; b) Based on the analysis of the next-to-leading order corrections to (6), which we are able to obtain (see (51)), we show that a genuine 1/N -expansion of the excess free energy for the 2D-OCP fails to exist if the fluid phase assumption is violated
Summary
In recent years there has been a considerable interest in the study of systems with logarithmic interactions. In the derivation of our results, it will be clear that: a) The fluctuations for x ≤ 1 are driven by a collective behaviour of the charges; on the contrary, the fluctuations to the right x > 1 are associated to a spontaneous symmetry breaking in the problem (the equilibrium configuration breaks the rotational invariance in the plane) This change of behaviour of the Coulomb gas is at the heart of the change of speed in the large deviations tails; b) Based on the analysis of the next-to-leading order corrections to (6), which we are able to obtain (see (51)), we show that a genuine 1/N -expansion of the excess free energy for the 2D-OCP fails to exist if the fluid phase assumption is violated (i.e. when the plasma distribution becomes singular).
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