EXACT FIELD-THEORETICAL DESCRIPTION OF PASSIVE SCALAR CONVECTION IN AN N-DIMENSIONAL LONG-RANGE VELOCITY-FIELD

We describe a new functional integral method for the computation of averages containing chronological exponentials of random matrices of arbitrary dimension. We apply these results to the rigorous study of the statistics of a passive scalar advected by a large-scale N-dimensional flow. In the delta-correlated case the statistics of the rate of line stretching appears to be exactly Gaussian at all times and we explicitly compute the dependence of the mean value and variance of the stretching rate on the space dimension N. The probability distribution function of the passive scalar is also exactly computed. Further applications of our functional integral method are suggested.