  Intermittency of $\mathcal{E}_N(x,g)=\exp\lbrack g| S_N(x)|^2\rbrack$ as
$N\to +\infty$ is investigated on a $d$-dimensional torus $\Lambda$, when
$S_N(x)$ is a finite Steinhaus series of $(2N+1)^d$ terms normalized to $<|
S_N(x)|^2> =1$. Assuming ergodicity of $\mathcal{E}_N(x,g)$ as $N\to +\infty$
in the domain $g<1$, where $\lim_{N\to +\infty}<\mathcal{E}_N(g)>$ exists,
transition to intermittency is proved as $g$ increases past the threshold
$g_{th}=1$. This transition goes together with a transition from (assumed)
ergodicity at $g<g_{th}$ to a regime where $\lim_{N\to
+\infty}\lbrack|\Lambda|<\mathcal{E}_N(g)>\rbrack^{-1}\int_{\Lambda}\mathcal{E}_N(x,g)
d^dx=0$ at $g>g_{th}$. In this asymptotic sense one can say that ergodicity is
lost as $g$ increases past the value $g=1$.
