  We revisit the results of Zamolodchikov and others on the deformation of
two-dimensional quantum field theory by the determinant $\det T$ of the stress
tensor, commonly referred to as $T\overline T$. Infinitesimally this is
equivalent to a random coordinate transformation, with a local action which is,
however, a total derivative and therefore gives a contribution only from
boundaries or nontrivial topology. We discuss in detail the examples of a
torus, a finite cylinder, a disk and a more general simply connected domain. In
all cases the partition function evolves according to a linear diffusion-type
equation, and the deformation may be viewed as a kind of random walk in moduli
space. We also discuss possible generalizations to higher dimensions.
