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Random Polymer Models

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Random Polymer Models, 9781860947865 (1860947867), Imperial College Press, 2007

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.”

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