Note on the Transition to Intermittency for the exponential of the Square of a Steinhaus Series

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 $g=1$.