Multiscaling in the Kraichnan Model of Passive Advection

$$ \mathcal{S}_{2\,n}(\boldsymbol{x};m)=\operatorname{E}|\theta(\boldsymbol{x};m)-\theta(\boldsymbol{0};m)|^{2\,n}\propto \|\boldsymbol{x}\|^{\zeta_{2\,n}(\xi)} $$ Kraichnan's model of passive advection is to-date the only PDE model of turbulence where intermittency and multiscaling can be proven in a mathematically controlled fashion. The result was first independently derived in the diffusion , large dimension , and laminar limits and numerically also by us here.
The connection between multiscaling and intermittency is due to the existence of statistical conservation laws, martingales, associated to shapes formed by Lagrangian particles in their many body translational invariant configuration space.
In retrospective, the result of Kupiainen et al. I am most pleased with, is having clarified the relation between quantum field theory methods and scaling predictions based on many body Hopf's equations.
Later here and here , I used these methods to exhibit how a geometric signature is encoded in the scaling exponent as their perturative expressions admits a classification in terms of Gel'fand-Zeitlin patterns of \(\mathrm{SL}(N\,,\mathbb{R})\) in agreement to an early general conjecture.