Exact extreme-value statistics at mixed-order transitions

We study extreme-value statistics for spatially extended models exhibiting mixed-order phase transitions (MOT). These are phase transitions that exhibit features common to both first-order (discontinuity of the order parameter) and second-order (diverging correlation length) transitions. We consider here the truncated inverse distance squared Ising model, which is a prototypical model exhibiting MOT, and study analytically the extreme-value statistics of the domain lengths The lengths of the domains are identically distributed random variables except for the global constraint that their sum equals the total system size L. In addition, the number of such domains is also a fluctuating variable, and not fixed. In the paramagnetic phase, we show that the distribution of the largest domain length l(max) converges, in the large L limit, to a Gumbel distribution. However, at the critical point (for a certain range of parameters) and in the ferromagnetic phase, we show that the fluctuations of l(max) are governed by novel distributions, which we compute exactly. Our main analytical results are verified by numerical simulations.