Fluctuations of the one-dimensional polynuclear growth model with external sources

The one-dimensional polynuclear growth model with external sources at edges is studied. The height fluctuation at the origin is known to be given by either the Gaussian, the GUE Tracy-Widom distribution, or certain distributions called GOE$^2$ and $F_0$, depending on the strength of the sources. We generalize these results and show that the scaling limit of the multi-point equal time height fluctuations of the model are described by the Fredholm determinant, of which the limiting kernel is explicitly obtained. In particular we obtain two new kernels, describing transitions between the above one-point distributions. One expresses the transition from the GOE$^2$ to the GUE Tracy-Widom distribution or to the Gaussian; the other the transition from $F_0$ to the Gaussian. The results specialized to the fluctuation at the origin are shown to be equivalent to the previously obtained ones via the Riemann-Hilbert method.