This work deals with a well-defined class of probabilistic models. The focus
is on polymers. More precisely I should say that the focus is on the
equilibrium statistical mechanics of a class of polymers: dynamical and
non-equilibrium phenomena are not treated, reflecting the fact that these
directions are at the moment under-developed, at least from the viewpoint
of the so called rigorous results. Moreover, only a subset of the world of
polymer models is taken into consideration.
If I try to characterize such a subset, keeping in mind the motivations
(that come from physics, biology, chemistry, material science, etc.), I end up
with a list: (1+d)-dimensional pinning models, (1+1)-dimensional wetting
models, adsorption models, copolymer models, DNA denaturation models,
etc. and to each model in this list one should probably add the adjective
disordered, since this work is mostly focused on disordered systems, even if
non-disordered systems do play a central role. But such a list may at first
appear quite disordered in itself...
If instead I take a purely mathematical standpoint, there is a totally
natural thread connecting the models in the list: take a (persistent or terminating)
renewal process and modify its law by giving rewards or penalties
at the renewal epochs, possibly depending on time lapsed since the previous
renewal epoch, and do that by introducing exponential, or Boltzmann,
weights. On page 685 of his celebrated review paperWalks, Walls, Wetting,
and Melting [Fisher (1984)] Michael E. Fisher writes:
“In fact, there is a rather simple but general mathematical mechanism
which underlies a broad class of exactly soluble one-dimensional models
which display phase transitions. This mechanism does not seem to be as
well appreciated as it merits and it operates in a number of applications we
wish to discuss.”