  Consider a bounded planar domain D, an instance h of the Gaussian free field
on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma <
2. The Liouville quantum gravity measure on D is the weak limit as epsilon
tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz,
where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h
on the circle of radius epsilon centered at z. Given a random (or
deterministic) subset X of D one can define the scaling dimension of X using
either Lebesgue measure or this random measure. We derive a general quadratic
relation between these two dimensions, which we view as a probabilistic
formulation of the KPZ relation from conformal field theory. We also present a
boundary analog of KPZ (for subsets of the boundary of D). We discuss the
connection between discrete and continuum quantum gravity and provide a
framework for understanding Euclidean scaling exponents via quantum gravity.
