Abstract
The free energy at zero temperature of Coulomb gas systems in generic dimension is considered as a function of a volume constraint. The transition between the ‘pulled’ and the ‘pushed’ phases is characterised as a third-order phase transition, in all dimensions and for a rather large class of isotropic potentials. This suggests that the critical behaviour of the free energy at the ‘pulled-to-pushed’ transition may be universal, i.e. to some extent independent of the dimension and the details of the pairwise interaction.
Highlights
Perhaps, the most popular third-order transitions are the so called Gross–Witten– Wadia [22, 40] and Douglas–Kazakov [14] large-N phase transitions
These models are naturally mapped onto the statistical physics of log-gases, and the strong-to-weak coupling transition of lattice gauge theories can be translated as a pulled-to-pushed transition for classical particles
Several other third-order phase transitions have been discovered over the last decade in many physics problems related at various levels to random matrices, such as the distribution of the maximum height of non-intersecting Brownian excursions [19, 38], conductance and shot noise in chaotic cavities [7, 10, 39], Rényi entanglement entropy of bipartite random pure states [13, 17, 35], wireless telecommunication [24], complexity of spin glass landscapes [20], and discrete log-gases related to random tilings [6]
Summary
Third-order phase transitions have been observed in one-dimensional and two-dimensional systems with logarithmic repulsion, i.e. for eigenvalues of unitarily invariant matrix models. In the GUE ensemble, the particles are constrained on the real line (n = 1); in the GinUE ensemble the particles live on the plane (n = 2) With this picture in mind, ZN in both cases is the partition function of a d = 2 Coulomb gas (− log |x| is the electrostatic potential in dimension two):. The form of the large deviation function is specific to the model (FGUE(R) = FGinUE(R)) In both cases the pulled-to-pushed transition is a third-order phase transition, i.e. the excess free energies are non-analytic with a discontinuous third derivative exactly at the critical point R = R :. Our findings below will instead show that a logarithmic repulsion is not essential to obtaining a third-order phase transition, and the source of this universality must be sought elsewhere
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