Random matrix theory is central to the study of the properties of strongly correlated systems in condensed matter physics and related fields, where important physical quantities are surprisingly well reflected by the joint probability density of the eigenvalues of random matrices. The mathematical expression for this density turns out to be formally linked to the physics of an ideal two-dimensional Coulomb gas of charged and interacting particles, influenced by an external potential V (x). The nature of V(x), together with the logarithmic interparticle potential, reflects the properties of the statistical ensemble chosen for the random matrices. For many choices, however, studies have found that the statistics of extremal eigenvalues follows a universal form – the so-called Tracy-Widom distribution – while the typical fluctuations of bulk eigenvalues scale logarithmically with the system size.
Previous studies indicate that variations of the dimensionality of the Coulomb gas may alter these results, giving distributions different from those of classic random matrix theory. In a new paper, LML Fellow Isaac Pérez Castillo and his students present a complete solution of the particle statistics of the one-dimensional case with an arbitrary external potential, obtaining exact expressions for both the typical and large-fluctuation regimes. They find that the typical fluctuations follow a universal function which depends solely on general properties of the external potential, and the asymptotic behaviour for extremal particles differs from the predictions of the Tracy-Widom distribution. Interestingly, the behaviour of bulk particles differs significantly from the previous studies. The analysis demonstrates that many of the paradigmatic properties of the full particle statistics of two-dimensional systems do not carry over to their one-dimensional counterparts. The one-dimensional Coulomb gas belongs to a different universality class.
The paper is available at https://arxiv.org/pdf/1803.11269.pdf