  In this paper, we prove the smooth cubic moments vanish for the Hecke--Maass
cusp forms, which gives a new case of the random wave conjecture. In fact, we
can prove a polynomial decay for the smooth cubic moments, while for the smooth
second moment (i.e. QUE) no rate of decay is known unconditionally for general
Hecke--Maass cusp forms. The proof bases on various estimates of moments of
central $L$-values. We prove the Lindel\"of on average bound for the first
moment of $\rm GL(3)\times GL(2)$ $L$-functions in short intervals of the
subconvexity strength length, and the convexity strength upper bound for the
mixed moment of $\rm GL(2)$ and the triple product $L$-functions. In
particular, we prove new subconvexity bounds of certain $\rm GL(3)\times GL(2)$
$L$-functions.
