Exact extremal statistics in the classical $1d$ Coulomb gas

Abstract

We consider a one-dimensional classical Coulomb gas of $N$ like-charges in a harmonic potential -- also known as the one-dimensional one-component plasma (1dOCP). We compute analytically the probability distribution of the position $x_{\max }$ of the rightmost charge in the limit of large $N$. We show that the typical fluctuations of $x_{\max }$ around its mean are described by a non-trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of $x_{\max }$ for the Dyson's log-gas. We also compute the large deviation functions of $x_{\max }$ explicitly and show that the system exhibits a third-order phase transition, as in the log-gas. Our theoretical predictions are verified numerically.

5 pages + 5 pages of supplementary material, 4 figures

References in corpus (10)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Growing interfaces uncover universal fluctuations behind scale invariance
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Large deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $\lambda H<0$
• A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix
• Universality of the third-order phase transition in the constrained Coulomb gas

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