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 ${\zeta }_n$ of the structure functions $S_n$ for $n\gg 1$ under the additional condition $d{\zeta }_2\gg 1$ (where $d$ is the dimensionality of space). At $n<n_c$ (where $n_c=d{\zeta }_2\mathrm{/}[2(2-{\zeta }_2)]$) the exponents are ${\zeta }_n=({\zeta }_2\mathrm{/}4)(2n-n^2\mathrm{/}{n}_c)$, while at $n>n_c$ they are $n$-independent: ${\zeta }_n={\zeta }_2{n}_c\mathrm{/}4$. We also estimate $n$-dependent factors in $S_n$, particularly their behavior at $n$ close to $n_c$.