Instanton for the Kraichnan passive scalar problem

We consider high-order correlation functions of the passive scalar in the Kraichnan model. Using the instanton formalism we find the scaling exponents ζ_n of the structure functions S_n for n>>1 under the additional condition dζ₂>>1 (where d is the dimensionality of space). At n<n_c [where n_c=dζ₂/2(2-ζ₂)] the exponents are ζ_n=(ζ₂/4)(2n-n²/n_c), while at n>n_c they are n independent: ζ_n=ζ₂n_c/4. We also estimate n-dependent factors in S_n, particularly their behavior at n close to n_c.