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We study the distribution of the position of the rightmost particle$x_{\max}$ in a $N$-particle Riesz gas in one dimension confined in a harmonictrap. The particles interact via long-range repulsive potential, of the form$r^{-k}$ with $-2<k<\infty$ where $r$ is the inter-particle distance. Inequilibrium at temperature $O(1)$, the gas settles on a finite length scale$L_N$ that depends on $N$ and $k$. We numerically observe that the typicalfluctuation of $y_{\max} = x_{\max}/L_N$ around its mean is of$O(N^{-\eta_k})$. Over this length scale, the distribution of the typicalfluctuations has a $N$ independent scaling form. We show that the exponent$\eta_k$ obtained from the Hessian theory predicts the scale of typicalfluctuations remarkably well. The distribution of atypical fluctuations to theleft and right of the mean $\langle y_{\max} \rangle$ are governed by the leftand right large deviation functions, respectively. We compute these largedeviation functions explicitly $\forall k>-2$. We also find that these largedeviation functions describe a pulled to pushed type phase transition asobserved in Dyson's log-gas ($k\to 0$) and $1d$ one component plasma ($k=-1$).Remarkably, we find that the phase transition remains $3^{\rm rd}$ order forthe entire regime. Our results demonstrate the striking universality of the$3^{\rm rd}$ order transition even in models that fall outside the paradigm ofCoulomb systems and the random matrix theory. We numerically verify ouranalytical expressions of the large deviation functions via Monte Carlosimulation using an importance sampling algorithm.

Edge fluctuations and third-order phase transition in harmonically confined long-range systems