  We study extreme value statistics (EVS) for spatially extended models
exhibiting mixed order phase transitions (MOT). These are phase transitions
which 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 (TIDSI) 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.
