Exact Computations in the Burgers Problem

We complete the program outlined in the paper of the author with A. Migdal and sum up exactly all the fluctuations around the instanton solution of the randomly large scale driven Burgers equation. The probability distribution coincides with the one conjectured by A. Polyakov within the applicability of the perturbation theory. In paper [1] we computed the probability distribution P (δu, r) of observing the velocity difference δu at a distance r for the randomly large scale driven Burgers equation in the WKB approximation. The results obtained there coincided with the one obtained earlier in [2]. Naturally, we were able to derive the probability distribution only for δu ≫ 0, the so-called “right tail” of P . We found P (δu, r) ≈ exp ( − 3 r ) (1) At the same time, the left tail, conjectured in [2] to be P (δu, r) ≈ r 3 2