This is joint work with Eyal Neuman.

Polymer models give rise to some of the most challenging problems in probability and statistical physics. For example, the typical end-to-end distance of a self-avoiding simple random walk is known only in one dimension and in dimensions greater than or equal to 5. The parameter \(n\) in the walk does not represent physical time, but rather the distance from one end of the polymer. There has been very little work on moving polymers, in spite of the obvious physical motivation.

We consider Funaki’s random string, which was also known to polymer scientists as the continuum limit of the Rouse model. Consider the stochastic heat equation with vector-valued solutions \(u(t,x)\in\mathbf{R}^d\) for \(x\in[0,J]\). Then \(t\) represents physical time and \(x\) is the length along the polymer, while \(u(t,x)\) is the position of the polymer. The SPDE is:

with Neumann boundary conditions, where \(\dot{W}\) is a \(d\)-dimensional vector of independent white noises. Next we impose an exponential weighting which penalizes self-intersection. In dimension \(d=1\), we study the radius \(R=R(t)\) of the polymer at a typical time \(t\), where \(R(t)\) measures the typical distance a point on the polymer from its center of mass.