Schrödinger equation driven by the square of a Gaussian field: instanton analysis in the large amplification limit

Abstract

We study the tail of p(U), the probability distribution of , for , being the solution to , where is a complex Gaussian random field, z and x respectively are the axial and transverse coordinates, with , and both and g > 0 are real parameters. We perform the first instanton analysis of the corresponding Martin-Siggia-Rose action, from which it is found that the realizations of S concentrate onto long filamentary instantons, as . The tail of p(U) is deduced from the statistics of the instantons. The value of g above which diverges coincides with the one obtained by the completely different approach developed in Mounaix et al (2006 Commun. Math. Phys. 264 741). Numerical simulations clearly show a statistical bias of S towards the instanton for the largest sampled values of . The high maxima—or ‘hot spots’—of for the biased realizations of S tend to cluster in the instanton region.