# Recent Progress on the Geometry of Random Walks

## 5th Peter Whittle colloquium

 Date: 25 May 2017 Location: Center for Mathematical Sciences Wilberforce Road Cambridge, CB3 0WB All lectures will take place in MR12. Program: 10.00 - 11.00. Artem Sapozhnikov 11.00 - 11.30. Coffee Break 11.30 - 12.30. Julien Poisat 12.30 - 15.00 Lunch 15.00 - 16.00. Amine Asselah 16.15 - 17.15. Frank den Hollander 17.30 - 18.30. Drinks reception REGISTRATION FORM (Please fill in by May 18) Probability seminar on 24 May, 4.15pm: Erwin Bolthausen Titles and abstracts Talk by Artem Sapozhnikov Random walk on a torus and random interlacements This is an overview talk about geometric properties of the simple random walk on a large discrete torus on time scales on which it visits a non-degenerate fraction of all the vertices. A considerable progress has been recently made by exploring connections between the random walk and the random interlacements. The geometry of the range (vertices visited by the walk) is by now well understood. The vacant set (unvisited vertices) undergoes a percolation phase transition, which makes its analysis more challenging, and a number of interesting open problems still stand. Incidentally, some are linked to open questions about the geometry of a simple random walk on the infinite integer lattice. Talk by Julien Poisat A toolbox for charged polymers In this talk I will give some details about the techniques that have been used up to now in the context of charged polymers (presented in Frank’s talk) as well as related models. One of the key tool is the Ray-Knight representation of local times for the one-dimensional model, which allows for a sharp representation of the partition function. (joint work with Frank den Hollander, Francesco Caravenna and Nicolas Pétrélis). I will also talk about what we can do in higher dimensions, where we use a comparison to weakly self avoiding walks, and the challenges to come. Talk by Amine Asselah Strong Law of Large Number for the capacity of a Wiener sausage in d=4 We show that the capacity of a Wiener sausage, scaled down by (t/log(t)), converges almost surely to 4 pi^2. In this talk, we hope to motivate the study of capacity of the sausage, and present a self-contained proof of the strong law, as well as some estimate for upward large deviations. This is a joint work with Bruno Schapira and Perla Sousi. Talk by Frank den Hollander Charged polymers Polymer chains carrying electric charges exhibit interesting scaling behaviour. Depending on the temperature and the distribution of the charges, they may undergo what is called a \emph{collapse transition}, between a phase in which they are spatially extended and a phase in which they are rolled up. This collapse transition can be seen as a simplified version of the \emph{folding transition} of a protein: interactions between different parts of the protein cause it to fold into different configurations depending on the temperature and the distribution of the amino acids. In this talk we consider a polymer chain on the $d$-dimensional integer lattice with random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The object of interest is the free energy per monomer in the limit as the length $n$ of the polymer chain tends to infinity. We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an \emph{extended phase} from a \emph{collapsed phase}. We identify the scaling properties of this critical curve, as well as the scaling properties of the polymer chain in each of the two phases. Both turn out to be anomalous. What makes the charged polymer model challenging is that the interaction is both attractive and repulsive. This places it outside the range of models that have been studied with the help of so-called sub-additivity techniques, and makes it into a testbed for the development of new approaches. Joint work with Quentin Berger (Paris), Francesco Caravenna (Milan), Dima Ioffe (Haifa), Nicolas Petrelis (Nantes), Julien Poisat (Paris). Organisers: Amine Asselah, Bruno Schapira, Perla Sousi