Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition

We study the distribution of the mean radial displacement of charges of a two-dimensional (2D) one-component plasma in the thermodynamic limit $N\to \infty$ at finite temperature $\beta >0$. We compute explicitly the large deviation functions showing the emergence of a fourth-order phase transition as a consequence of a change of topology in the plasma distribution. This weak phase transition occurs exactly at the ground state of the plasma. These results have been compared with the integrable case (finite $N$) of plasma parameter $\beta q^2=2$. In this case the problem can be mapped to the stationary properties of 2D Dyson Brownian particles and to a non-Hermitian matrix model.

Figure 1

Top: Rate function $\mathrm{\Psi }\left(x\right)$ from Eq. (20) and its leading terms for $x\to 0$, $x\simeq 2/3$ and $x\to \infty$ [see Eq. (7)]. At $x=\langle {\mathrm{\Delta }}_N\rangle \equiv 2/3$ the function is not analytic. Central: Support $\mathrm{\Sigma }\left(s\right)$ of the equilibrium distribution ${\mu }_s^{☆}$ of the plasma for different values of $s$. From left to right $s=1,0,-0.5$. Bottom: The corresponding radial distributions ${\int }_0^{2\pi }d{\mu }_s^{☆}\left(\right|\vec{r}|,\phi )$.

Figure 2

Top: The cumulant generating function $J\left(s\right)$ in Eq. (5) (solid line) compared with the exact finite $N$ formula (21) (at $\beta =2$) for $N=16$ (dots). Already for such a small value of $N$, the large deviation function $J\left(s\right)$ matches the finite $N$ expression of $-1/\left(2N^2\right)log{\widehat{\mathcal{P}}}_{N,\beta =2}\left(s\right)$ with an error $\mathcal{O}\left({10}^{-2}\right)$ in the plotted region. Bottom: Rescaled cumulants $N^{2(r-1)}{\kappa }_m\left({\mathrm{\Delta }}_N\right)$ for $m=1,2,3,4$ at increasing values of $N$ for $\beta =2$. The first three cumulants have a regular behavior in $N$ and approach their leading order (6), while $N^6{\kappa }_4$ behaves irregularly and does not converge for large $N$.