  We give a new proof of the fact that the solutions of the stochastic heat
equation, started with non-negative initial conditions, are strictly positive
at positive times. The proof uses concentration of measure arguments for
discrete directed polymers in Gaussian environments, originated in M.
Talagrand's work on spin glasses and brought to directed polymers by Ph.
Carmona and Y. Hu. We also get slightly improved bounds on the lower tail of
the solutions of the stochastic heat equation started with a delta initial
condition.
