Absence of first order transition in the random crystal field Blume-Capel model on a fully connected graph

In this paper we solve the Blume-Capel model on a complete graph in the presence of random crystal field with a distribution, $P(\Delta_i) =p\delta(\Delta_i-\Delta)+(1-p) \delta(\Delta_i+\Delta)$, using large deviation techniques. We find that the first order transition of the pure system is destroyed for $0.046<p<0.954$ for all values of the crystal field, $\mathrm{\Delta }$. The system has a line of continuous transition for this range of $p$ from $-\mathrm{\infty }<\mathrm{\Delta }<\mathrm{\infty }$. For values of $p$ outside this interval, the phase diagram of the system is similar to the pure model, with a tricritical point separating the line of first order and continuous transitions. We find that in this regime, the order vanishes for large $\mathrm{\Delta }$ for $p<0.046$(and for large $-\mathrm{\Delta }$ for $p>0.954$) even at zero temperature.