Decay of Metastable Vacuum in Liouville Gravity

A decay of weakly metastable phase coupled to two-dimensional Liouville gravity is considered in the semiclassical approximation. The process is governed by the ``critical swelling'', where the droplet fluctuation favors a gravitational inflation inside the region of lower energy phase. This geometrical effect modifies the standard exponential suppression of the decay rate, substituting it with a power one, with the exponent becoming very large in the semiclassical regime. This result is compared with the power-like behavior of the discontinuity in the specific energy of the dynamical lattice Ising model. The last problem is far from being semiclassical, and the corresponding exponent was found to be 3/2. This exponent is expected to govern any gravitational decay into a vacuum without massless excitations. We conjecture also an exact relation between the exponent in this power-law suppression and the central charge of the stable phase.