Directed polymer inside a parabola: Exact solution

We consider a directed polymer on a square lattice confined inside a parabola with boundaries Y=±${\mathit{Cx}}^{\mathrm{\alpha }}$. Generalizing standard transfer-matrix methods, an effective transfer operator was introduced that takes into account slow variation of the size of the system in the y direction. By an exact calculation the effect of the confinement is found to be relevant and irrelevant for α1/2, respectively; in the former case the thermodynamical functions have a stretched-exponential behavior around the critical point. For α=1/2 the geometrical constraint is marginal, and the critical exponents are nonuniversal.