DIRECTED POLYMER INSIDE A PARABOLA - EXACT SOLUTION

We consider a directed polymer on a square lattice confined inside a parabola with boundaries Y = +/- Cx(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 alpha < 1/2 and > 1/2 , respectively; in the former case the thermodynamical functions have a stretched-exponential behavior around the critical point. For alpha = 1/2 the geometrical constraint is marginal, and the critical exponents are nonuniversal.