We study two problems related by a common set of techniques. In the first problem, we consider a model for the distribution of a long homopolymer in a potential field. For various values of the temperature, including those at or near the critical value, we consider the limiting behavior of the polymer when its size tends to infinity.

In the second problem, we investigate the long-time evolution of branching diffusion processes in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. In the super-critical case, we describe the asymptotics of the number of particles in a given domain and describe the growth of the region containing the particles. In the sub-critical regime, we describe the limiting distribution of the total number of particles.