appear multiplied by $%
g^{\prime }$, seem to play a very important role for the strength of the EWPT. Therefore we focus on these parameters while fixing $g^{\prime
}=g_{1},~m_{Q}=m_{U}=1~TeV,$ and choosing different values for $A_{t},\
M_{2},\ M_{1}$, and $M_{1}^{\prime }$. We will allow the parameters $\lambda
,\ \upsilon _{x},\ Q_{1,2},\ A_{\lambda }$, and $\theta _{0}$ to vary randomly within the intervals :$$\begin{array}{ccc}
0.001<\lambda <0.5, &  & 0.5<\upsilon _{x}/TeV<4, \\
1<\tan \beta <20, &  & -1.2<A_{\lambda }/TeV<1.2, \\
-4<Q_{1,2}<4, &  & 0<\theta _{0}<\pi /3.%
\end{array}
\label{pra}$$

The electroweak phase transition
================================

Due the condition (\[zmix\]) and (\[zpm\]), there could exist a hierarchy between the vev of the singlet and those of the doublets, i.e., $%
\upsilon _{x}>>\upsilon _{1,2}$. In this case, the gauge symmetry could be broken in two step. However, in the case where the mixing is extremely suppressed as in region **(1)** (\[zmixing\]), the singlet vev could be low as $\sim 500$ $GeV$. In this case, the gauge symmetry could be broken just in one step. In the two-step symmetry breaking case, one notices that above a certain high temperature, the singlet vev was zero $\upsilon
_{x}\left( T_{>>}\right) =0$. This can be seen by putting $\upsilon _{1,2}=0$ in (\[V1\]), 