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

Intermittency of ${\mathcal{E}}_N(x,g)=\exp [g\mathrm{\mid }{S}_N(x){\mathrm{\mid }}^2]$ as $N\to +\mathrm{\infty }$ is investigated on a $d$-dimensional torus $\mathrm{\Lambda }$, when $S_N(x)$ is a finite Steinhaus series of $(2N+1{)}^d$ terms normalized to $<| S_N(x)|^2> =1$\mathrm{.}Assumingergodicityof$$\mathcal{E}$N(x,g)$as$N\to +\infty$inthedomain$g<1$,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 $g=1$.