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 xmax of the rightmost charge in the limit of large N . We show that the typical fluctuations of xmax around its mean are described by a non-trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of xmax for the Dyson's log-gas. We also compute the large deviation functions of xmax explicitly and show that the system exhibits a third-order phase transition, as in the log-gas. Our theoretical predictions are verified numerically.The Tracy-Widom (TW) distribution has emerged ubiquitously in diverse systems in the recent past [1, 2]. It was originally discovered as the limiting distribution of the top eigenvalue x max of an N × N Gaussian random matrix [3]. Since then, it has appeared in various areas of physics [4,5], mathematics [6,7], and information theory [8]. For example, in physics it has appeared in stochastic growth models and related directed polymer in 1 + 1 dimensional random media belonging to the Kardar-Parisi-Zhang (KPZ) universality class [9-15], non-intersecting Brownian motions [16], non-interacting fermions in a one-dimensional trapping potential [17][18][19], disordered mesoscopic systems [20] and even in the YangMills gauge theory in 2-dimensions [16]. It has also been measured experimentally in several systems including liquid crystals [21], coupled fiber lasers [22] or disordered superconductors [23]. The TW distribution describes the probability of typical fluctuations of x max around its mean. In contrast, the atypical fluctuations of x max to the left and right, far from its mean, are described respectively by the left and right large deviation tails. These tails have been computed explicitly [24][25][26][27] and shown to correspond to two different thermodynamic phases separated by a third order phase transition [28,29]. Similar third order phase transitions have also been found in a variety of other systems [29][30][31][32][33][34][35].For Gaussian ensembles in random matrix theory (RMT), the joint probability distribution function (PDF) of the N real eigenvalues {x 1 , · · · , x N } is known explicitly [36,37] P({x i }) = B N e

Phase transitions and Wilson loops in antisymmetric representations in Chern–Simons-matter theory

“…It would be interesting to explicitly check this, by adding a FI parameter in [25] and doing the corresponding analysis. Finally, it is worth mentioning that at first apparently unrelated works in statistical mechanics, study in fact similar problems and systems [29,30]. and the saddle point equation reduces to the trivial expression f ≡ 0.…”

Complex (super)-matrix models with external sources and q-ensembles of Chern–Simons and ABJ(M) type

“…What it gives is the probability of an atypically large fluctuation, with the greatest eigenvalue moving deep into the bulk of the eigenvalue distribution. This is a q-analogue of the large fluctuations in the GUE discussed in [24,68] (see also [69,70]). Besides, we underline that, differently from [24] where a large deviation from the equilibrium configuration is of order ∼ √ N , in the q-analogue the support of the eigenvalue density grows as N g s , thus large deviations from the equilibrium are of order ∼ N .…”

Some Connections Between the Classical Calogero–Moser Model and the Log-Gas

“…An example of an interesting question that one could ask is regarding the distribution of the edge particle which is known to be of the Tracy-Widom form for the log-gas. In a recent paper the question of universality of this result for general 1D interacting systems was investigated and it was shown that the form is in fact very different for a 1D coulomb gas (interaction potential between particles given by modulus of distance) [35]. In the present paper we ask the same question for the Calogero-Moser model.…”