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

Abstract

Intermittency of ${\mathcal{E}}_N(x,g)=\exp [g|S_N(x){|}^2]$ 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 ${\lim }_{N\to +\infty }[|\Lambda |<{\mathcal{E}}_N(g)>{]}^{-1}{\int }_{\Lambda }{\mathcal{E}}_N(x,g)d^{d}x=0$ at $g>g_{th}$. In this asymptotic sense one can say that ergodicity is lost as $g$ increases past the value $g=1$.

REVTeX file, 12 pages, changed introduction and added references, accepted by J. Phys. A

References in corpus (1)

• Erratum: Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrodinger Equation Driven by the Square of a Gaussian Field