There is a beautiful relation between the DP model and the dynamics of a growing interface. To see this, we consider the restricted partition function,
This restricted partition sum is proportional to the probability that the DP's endpoint (at internal coordinate x) is located at position . Eq. 124 is nothing else but the path-integral formulation of the imaginary-time Green's function of a particle in a time-dependent potential , where is the coordinate of the particle, x is the imaginary time, and T plays the role of . It therefore satisfies the Schrödinger equation
It is very natural to consider the free energy
which thus obeys the equation
where , , and . Eq. 127 is the celebrate Kardar-Parisi-Zhang (KPZ) equation describing the height profile of a growing surface. This mapping allows us to relate the height fluctuations of this surface to free energy fluctuations of the DP. The self-affine roughness of the polymer corresponds to dynamical scaling in the interface dynamics. Indeed, an extremely detailed and rich analogy can be made, which allows for various approximate calculations of properties for the pinned DP.