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

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