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# YEP X “Random Polymers”

## Jan 14, 2013, 08:00 - Jan 18, 2013, 17:00

#### Summary

The theme of the workshop is random polymers in the broad sense of the word. Targeted are not only researchers from probability theory, but also from combinatorics, physics and physical chemistry. Polymers with disorder are part of the workshop, but are not the main focus.

#### Sponsors

#### Organizers

Frank den Hollander | University of Leiden |

Vladas Sidoravicius | Universidade Federal do Rio de Janeiro |

Stu Whittington | University of Toronto |

#### Speakers

Ken Alexander | University of South California |

Matthias Birkner | University of Mainz |

Mireille Bousquet-Mélou | CNRS LaBRI Université Bordeaux 1 |

Richard Brak | University of Melbourne |

Philippe Carmona | Université de Nantes |

Dimitris Cheliotis | University of Athens |

Xia Chen | University of Tenessee |

Francis Comets | Université Paris-Diderot Paris 7 |

Ivan Corwin | New York University |

Sacha Friedl | Universidade Federal de Minas Gerais |

Giambattista Giacomin | Université Paris Diderot |

Tony Guttmann | University of Melbourne |

Gary Iliev | York University |

Dima Ioffe | Technion-Israel Institute of Technology |

Hubert Lacoin | University Paris Dauphine |

Fabio Martinelli | University of Roma Tre |

Alex Opoku | University of Leiden |

Aleks Owczarek | University of Melbourne |

Nicolas Pétrélis | Université de Nantes |

Thomas Prellberg | Queen Mary University of London |

Buks van Rensburg | York University |

Chris Soteros | University of Saskatchewan |

Rongfeng Sun | National University of Singapore |

Fabio Toninelli | Laboratoire de Physique ENS Lyon |

Carlo Vanderzande | Universiteit Hasselt |

Nobuo Yoshida | Kyoto University |

Nikos Zygouras | Warwick University |

#### Abstracts

**Path properties of the random pinning polymer in the delocalized regime
**We study the path properties of the random pinning model in the delocalized regime and we prove that, at any temperature, the likely number of contacts with the defect line under the (random) Gibbs measure stays bounded in probability as the polymer length grows. On the other hand we also show that at sufficiently low temperature, there exists a.s. a subsequence where the likely number of contacts grows like the log of the length of the polymer.

(joint work with Nikos Zygouras)

**Quenched large deviations for continuous words and an application to the Brownian copolymer
**We describe the quenched large deviation rate function for the empirical processes obtained by cutting a Brownian path along a continuous-time renewal process, and discuss how this can be employed to study a continuous copolymer model.

(joint work in progress with Frank den Hollander)

**The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1+\sqrt 2
**In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal lattice is \mu=\sqrt{2+\sqrt 2}. A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with n in [-2,2] (the case n=0 corresponding to SAWs). We modify this model by restricting to a half-plane and introducing a fugacity associated with boundary sites (also called surface sites) and obtain a generalisation of Smirnov’s identity. The value of the critical surface fugacity was conjectured by Batchelor and Yung in 1995. This value also plays a crucial role in our identity, which thus provides an independent prediction for it. For the case n=0, corresponding to SAWs interacting with a surface, we prove the conjectured value of the critical surface fugacity. A critical part of this proof involves demonstrating that the generating function of self-avoiding bridges of height T, taken at its critical point 1/\mu, tends to 0 as T increases, as predicted from SLE theory.

(joint work with Nicholas R. Beaton, Jan de Gier, Hugo Duminil-Copin and Anthony J. Guttmann)

**Polymers, Particle Hopping and a change in basis
**This talk will discuss the connection between an equilibrium polymer phase transition and the non-equilibrium phase transition in a particle hopping model and how this connection arouse out of studying the combinatorics of the change in vector space basis of ASEP algebra representations.

**Phase transition for the partially directed polymer
**A long linear polymer in a dilute solution is modelled by a partially directed self avoiding random walk with self-interactions. We prove the existence of a collapse transition, identify the critical temperature, and establish that the order of the transition is 3/2

**Exponential asymptotics for time-space Hamiltonians**

**On the quenched critical curve of the random pinning model
**We improve a recent formula for the quenched critical curve of the random pinning model, and show some of its applications.

**Mean-field Last Passage Percolation
**We consider a system of N particles with a stochastic dynamics introduced by Brunet and Derrida. The particles can be interpreted as last passage times in directed percolation on {1,…,N} of mean-field type. The particles remain grouped and move like a traveling wave, subject to discretization and driven by a random noise. As N increases, we obtain estimates for the speed of the front and its profile, for different laws of the driving noise. The Gumbel distribution plays a central role for the particle jumps, and we show that the scaling limit is a Levy process in this case. The case of bounded jumps yields a completely different behavior.

(joint work with J. Quastel, A. Ramirez)

**A rigorous replica trick for directed polymers
**We develop a rigorous replica trick for certain integrable deformations of the directed polymer model. Under the limit which recovers the polymer, the rigorous approach degenerates to the physicists famously non-rigorous manipulations. ASEP and q-TASEP both work as deformations and inspiration from the theory of Macdonald processes serves as the key to the computations.

(joint work withAlexei Borodin and Tomohiro Sasamoto)

**Chains with complete connections and modified majority rules
**We take a closer look at a type of chains with complete connections introduced by Berger, Hoffman and Sidoravicius [BHS]. In that model, the distribution of the present when conditioned on the whole past is determined by the majority rule, looking at the values of the process on a random finite subset of the past.

We present the general ideas behind the [BHS]-mechanism, from a random environment perspective. As shown in [BHS], with a pure majority rule, relevant information travels from the remote past to the remote future with positive probability (with respect to the environment). We prove this by mapping the effective multiscale model to a one-dimensional non-homogeneous Isong model, in which the nature of the transition is transparent.

On the other hand, we show that when the majority rule is modified so as to be differentiable at the origin, uniqueness always holds (with probability one with respect to the environment).

(joint work with J.C.A. Dias (UFOP, Ouro Preto))

**Oscillatory critical amplitudes and near-constancy phenomena
**The talk focuses on hierarchical pinning models, that is pinning models on diamond lattices, and aims at analyzing the critical behavior. An unexpected phenomenon takes place: the critical behavior is not ‘pure power law’, in the sense that the power law has a prefactor – the amplitude – which is not constant, but periodic. The aim of the talk is in particular to show the tight link with an analogous phenomenon observed for Galton-Watson trees by T. E. Harris (and in several other branches of mathematics after that) and to explain why this oscillation is directly linked to the geometry of the Julia set of the map that defines the diamond lattice (or the Galton-Watson tree): this link had been conjectured by Derrida, Itzykson and Luck in 1984. Other recent results will be reported.

**Off-critical parafermions and the winding angle distribution of SAWs
**Using an off -critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents and wedge critical exponents with the exponent characterising the winding angle distribution of the SAW model in the half-plane, or more generally in a wedge. We assume only the existence of these exponents. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments.

**Pattern recognition in polymeric systems: modeling copolymers near an inhomogeneous surface
**We present partially-directed and self-avoiding walk models of copolymers interacting with an inhomogeneous surface, in the absence and presence of an elongational force. In all cases, we observe qualitative agreement between the results obtained via the directed walk model and those from self-avoiding walks. When the polymer is subject to an elongational force, we also use a simple low temperature treatment to understand the behaviour of both models. We solve the directed walk problem essentially completely and study the force-temeprature phase diagram in various regimes of the direction and strength of the applied force.

(joint work with S. G. Whittington and E. Orlandini)

**Dima Ioffe**

**Interaction, repulsion and scaling of low temperature interfaces
**I shall describe recent results on interaction and entropic repulsion between low temperature Ising type interfaces (which, given the circumstances, will be called Ising polymers) in two dimensions.

(joint work with Senya Shlosman, Fabio Toninelli and Yvan Velenik)

**Counting self-avoiding paths on an infinite supercritical percolation cluster
**The self-avoiding walk on $\mathbb{Z}^d$ has been introduced by Flory and Ott as a natural model for polymers. In spite of the apparent simplicity of the model, mathematicians understanding is very far of being complete, in particular in low dimension (d=2,3,4). For this reason the disordered version of the model: Self-avoiding walk in a random potential has not received much attention from the mathematical community. On the other hand the model has received some attention in the Physics literature and some conjectures have been formulated.

Our aim is to approach the problem by studying the asymptotic of the partition function. A particular case of interest is the one where the environment is given by supercritical Bernouilli percolation.

We obtained so far two results: that when $d=2$ the model is is never self-averaging even for small dilution in the sense that the number of open path is typically exponentially smaller than its average, and that the same phenomenon occur just above the percolation threshold in high dimension.

**Dynamics of 2 + 1 dimensional sos surfaces above a wall: slow mixing induced by entropic repulsion
**We study the Glauber dynamics for the (2+1)d Solid-On-Solid model above a hard wall and below a far away ceiling, on an LxL box of Z^2 with zero boundary conditions, at large inverse-temperature β.

Reference arXiv:1205.6884

**Alex Opoku**

**Copolymer with Pinning: A view from a variational window
**In a recent paper, Dimitris Cheliotis and Frank den Hollander have developed a variational approach to the study of the random pinning model, based on a large deviation principle for random words drown from random letters by M. Birkner, A. Greven and F. den Hollander.

The goal of this talk is to highlight the potential of this variational approach by applying it to the random copolymer with pinning model. In particular, we identify the structure of the phase diagram.

(joint work with Erwin Bolthausen and Frank den Hollander)

**Exact solution of two friendly walks above a sticky wall with single and double interactions
**We find, and analyse, the exact solution of two friendly directed walks, modelling polymers, which interact with a wall via contact interactions. We specifically consider two walks that begin and end together so as to imitate a polygon. We examine a general model in which a separate interaction parameter is assigned to configurations where both polymers touch the wall simultaneously, and investigate the effect this parameter has on the integrability of the problem. We find an exact solution of the generating function of the model, and provide a full analysis of the phase diagram that admits three phases with one first-order and two second-order transition lines between these phases.

**Phase diagram of a copolymer in an emulsion
**In this talk we consider a model for a random copolymer, made of monomers of type A and B, immersed in a micro-emulsion of random droplets of type A and B. The emulsion is modeled by large square blocks filled randomly with solvents A and B and the possible configurations of the copolymer are given by the trajectories of a partially directed random walk in dimension 2. The copolymer and the emulsion interact via an Hamiltonian which favors matches and disfavors mismatches.

In a similar model that has been studied recently, some restrictions were imposed to the random walk trajectories and a variational formula was derived for the quenched free energy per monomer.

In the present model we drop almost all those path restrictions and we display a new variational formula for the free energy. The latter formula involves the fractions of time the copolymer moves at a given slope through the interior of solvents A and B and the fraction of time it moves along AB-interfaces.

With the help of this variational formula, we manage to give the general structure of the phase diagram in both the supercritical regime (A blocks percolate) and the subcritical regime ($A$ blocks do not percolate).

**The pressure of surface-attached polymers and vesicles
**A polymer grafted to a surface exerts pressure on the substrate. Similarly, a surface-attached vesicle exerts pressure on the substrate. By using directed walk models, we compute the pressure exerted on the surface for grafted polymers and vesicles, and the effect of surface binding strength and osmotic pressure on this pressure.

**Some results on inhomogeneous percolation
**Abstract

**Knot transition probabilities and knot reduction for lattice strand passage models of ring polymers**

**Symmetric Rearrangements Around Infinity with Applications to Levy Processes
**We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger and can be interpreted as involving symmetric rearrangements of domains around infinity. As applications, we prove two comparison results for general Levy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Levy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Levy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi for the Wiener sausage. In the second application, we show that the q-capacity of a Borel measurable set for a Levy process can only increase if the set and the Levy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe for symmetric Levy processes.

(joint work wiht Alexander Drewitz and Perla Sousi)

**L^{1/3} fluctuations for the contours of the 2D SOS model
**We consider the two-dimensional discrete SOS model at low temperature T in a LxL box. In presence of a wall at height zero, the surface is tipically at height (T/4)log(L) (due to entropic repulsion). We show that the level lines of the interface, once rescaled by 1/L, have a deterministic limit described by a suitable Wulff shape, and that the fluctuations around the limit shape are of order L^{1/3} (before rescaling). This seems to be connected with the “1/3-type fluctuations” one finds in growth models and directed polymers in (1+1) dimensions.

(joint work with P. Caputo, E. Lubetzky, F. Martinelli and A. Sly)

**Fractional Brownian motion and the critical dynamics of zipping polymers
**We consider two complementary polymer strands of length L attached by a common end monomer. The two strands bind through complementary monomers and at low temperatures form a double stranded conformation (zipping), while at high temperature they dissociate (unzipping).

This is a simple model of DNA (or RNA) hairpin formation. Here we investigate the dynamics of the strands at the equilibrium critical temperature using Monte Carlo Rouse dynamics. We find that the dynamics is anomalous, with a characteristic time scaling as $\tau \sim L^{2.26(2)}$, exceeding the Rouse time $\sim L^{2.18}$. We investigate the probability distribution function, the velocity autocorrelation function, the survival probability and boundary behavior of the underlying stochastic process. These quantities scale as expected from a fractional Brownian motion with a Hurst exponent $H=0.44(1)$. We discuss similarities and differences with unbiased polymer translocation.

**Brownian Directed Polymers in Random Environment: Complete Localization and Phase Diagram
**This is a joint work with Francis Comets. We study a model of directed polymers in random environment in dimension $1+d$, givenby a Brownian motion in a Poissonian potential. We study the effect of the density and the strength of inhomogeneities, respectively the intensity parameter $\nu$ of the Poisson field and the temperature inverse $\beta$. Our results are: (i) fine information on the phase diagram, with quantitative estimates on the critical curve; (ii) pathwise localization at low temperature and/or large density; (iii) complete localization in a favourite corridor for large $\nu \beta^2$ and bounded $\beta$.

**Geometric RSK, Whittaker Functions and Random Polymers (with or without pinning)
**Whittaker functions are special functions, which have a central position in representation theory and integrable systems. Surprisingly, they turn out to play a central role in the fluctuation analysis of Random Polymer Models. In this talk I will explain the emergence of Whittaker functions in Random Polymer Models, via the use of Geometric Robinson-Schensted-Knuth correspondence (RSK). I will then describe their role in the computation of the distribution of partition functions, where also a certain pinning may be allowed.

(joint work with I.Corwin, N. O’Connell and T.Sepälläinen)