Final Report Summary - MORAMERAPO (Motion in Random Media: Random Polymers)
The project has focused on the study of models of motion in random media. Within this class, particular attention has been paid on "random polymer models".
It is a classic result in probability theory that a simple random walk will be at distance, which is proportional to the square root of the time it has run. However, if the walk runs in a disordered medium, this diffusive behaviour can drastically change, leading to rich physical phenomena. Within, this project we have worked on classifying and characterising the effect of disorder on the behaviour of random walks, identifying phase transitions and new universality classes. Our work lied on three pillars: (A) Disorder on a Defect Line (pinning and copolymer models), (B) Bulk-Disorder Directed Polymers and (C) Bulk-Disorder Semidirected Polymers. The main outcomes of our research are summarised as follows:
(A) Pinning and copolymer models are used extensively to model physical and biophysical phenomena. They exhibit an interesting transition between a localised and a delocalised phase and the identification of their phase diagram has motivated the building of several rigorous and non rigorous methods.
In publication 5. we succeeded in identifying completely the phase diagram at small coupling constants (i.e. close to the origin), managing to even identify the asymptotic constant. This is the sharpest result in the area and although currently restricted to the case where the underlying process is positive recurrent, there is hope of extension to the more general case of processes in the domain of attraction of stable laws.
In publication 4, we managed to resolve the question of whether in the delocalised regime, the random polymer has finite number of contact points with the defect line. That was a question which had remained open since the first mathematical treatments of this class of models, about ten years ago. We showed that, in probability, there is indeed a finite number of contacts, while almost surely there are atypical sequences of disorder, that will induce a logarithmic number of contacts.
In publication 3. we initiated a programme of establishing continuum limits of disordered systems. Starting from the pinning partition function, we were led to a more general setting, that included other disordered models such the newly introduced "long-range directed polymer" and Random Field Ising model. A byproduct of our work was the establishment of a new framework of "disorder relevance". This is a concept introduced in the 70's by Harris in the physics literature, attempting to classify systems where the introduction of disorder drastically changes the phase diagrams. Furthermore, in a preprint entitled "The continuum pinning model" (joint with F. Caravenna and R.Sun) we extend our analysis to construct the continuum pinning measure and establish its fine properties. Finally, we have in the course (jointly again with Caravenna and Sun) of establishing continuum limits at marginal relevance. We have been developing new concepts of limit theory, that promise to reveal new universality features among large classes of models that exhibit marginal relevance.
(B) Directed Polymer models belong to the Kardar-Parisi-Zhang universality class. In the mid 80's, Kardar-Parisi-Zhang proposed a nonlinear stochastic PDE (KPZ equation) to describe the fluctuations of randomly growing surfaces. Although highly ill posed, this equation was used to predict a new universality class, where the fluctuations are characterised by an exponent 1/3, instead of the usual 1/2 of the central limit theorem. The directed polymer model can be considered as a discretisation of the solution of the KPZ equation providing a route to test the 1/3 prediction. However, not much progress had been made in the analysis of the directed polymer model. A reason for this was that its "integrable" structure had not been understood.
In publications 1 and 2 we succeeded in making a leap in the understanding of the structure of the model. Bringing in ideas from Tropical Combinatorics, Representation theory (Robinson-Schensted-Knuth algorithm) and Whittaker functions, we managed to identify the Laplace transform of a directed polymer with log-gamma disorder. This work was then the starting point in the "Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity". Comm. Math. Phys.,324:215–232 (2013), by Borodin-Corwin-Remenik, which established the 1/3 exponent and the Tracy-Widom limit theorem.
Our works ,  have opened new pathways to understand the KPZ universality and connect with other areas of mathematics. We are in the course of further elaborating in this direction.
(C) Semidirected polymers mimic more realistically polymer chains, as the models are not any more directed, but allow for backtracks.
Although many results from the study of directed models are expected to be valid in the non directed setting, the technical difficulties are substantial, not allowing for significant progress.
In publication 6. we managed to transfer ideas from the directed setting, in order to establish the existence of a "strong disorder regime". This regime can be characterised by the inequality between the annealed and quenched free energies. Furthemore, we showed that in the strong disorder regime the polymer path localises, which is in contrast to a situation of a diffusive setting.
In publication 7, using ideas from concentration of measure we established that the fluctuations of the partition function are subgaussian. Moreover, using detailed coarse graining analysis we obtained subgaussian rates of convergence of the logarithm of the partition function to the free energy.
The type of ideas developed in these works are expected to be useful in the future for establishing fluctuation estimates in high dimensional polymers or in general non solvable cases.