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We consider the solution of the stochastic heat equation \partial_T \mathcal{Z} = 1/2 \partial_X^2 \mathcal{Z} - \mathcal{Z} \dot{\mathscr{W}} with delta function initial condition \mathcal{Z} (T=0)= \delta_0 whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions.
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