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

International audienceWe study the distribution of the position of the rightmost particle x $_{max}$ in a N-particle Riesz gas in one dimension confined in a harmonic trap. The particles interact via long-range repulsive potential, of the form r $^{−k}$ with −2 −2. We also find that these LDFs describe a pulled to pushed type phase transition as observed in Dyson’s log-gas (k → 0) and 1d one component plasma (k = −1). Remarkably, we find that the phase transition remains third order for the entire regime. Our results demonstrate the striking universality of the third order transition even in models that fall outside the paradigm of Coulomb systems and the random matrix theory. We numerically verify our analytical expressions of the LDFs via Monte Carlo simulation using an importance sampling algorithm