Technische Universiteit Eindhoven

One of the holy grails of material science is a complete characterization of ground states of material energies. Some materials have periodic ground states, others have quasi-periodic states, and yet others form amorphic, random structures. Knowing this structure is essential to determine the macroscopic material properties of the material. In theory the energy contains all the information needed to determine the structure of ground states, but in practice it is extremely hard to extract this information.

In this talk I will describe a model for which we recently managed to characterize the ground state in a very complete way. The energy describes the behaviour of diblock copolymers, polymers that consist of two parts that repel each other. At low temperature such polymers organize themselves in complex microstructures at microscopic scales.

We concentrate on a regime in which the two parts are of strongly different sizes. In this regime we can completely characterize ground states, and even show stability of the ground state to small energy perturbations.

This is work with David Bourne and Florian Theil.

Weakly self-avoiding walk (WSAW) is obtained by giving a penalty for every

self-intersection to the simple random walk path. The Edwards model (EM) is

obtained by giving a penalty proportional to the square integral of the local

times to the Brownian motion path. Both measures significantly reduce the

amount of time the motion spends in self-intersections.

The above models serve as caricature models for polymers, and we will give

an introduction polymers and probabilistic polymer models. We study the WSAW

and EM in dimension one.

We prove that as the self-repellence penalty tends to zero, the large

deviation rate function of the weakly self-avoiding walk converges to the rate

function of the Edwards model. This shows that the speeds of one-dimensional

weakly self-avoiding walk (if it exists) converges to the speed of the Edwards

model. The results generalize results earlier proved only for nearest-neighbor

simple random walks via an entirely different, and significantly more

complicated, method. The proof only uses weak convergence together with

properties of the Edwards model, avoiding the rather heavy functional analysis

that was used previously.

The method of proof is quite flexible, and also applies to various related

settings, such as the strictly self-avoiding case with diverging variance.

This result proves a conjecture by Aldous from 1986. This is joint work with

Frank den Hollander and Wolfgang Koenig.