Workshop on

**Random Polymers**

**June 18-22, 2007**

**EURANDOM, Eindhoven, The Netherlands**

**K. Alexander - University of Southern California
**Joint work with Nikos Zygouras

**Distinct quenched and annealed critical points in polymer depinning
transitions **

We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has a "barely infinite" mean, with the probability of an excursion of length n decaying like n^{-2}g(n) with g slowly varying. We show that for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.

**
J. Alvarez -University of Toronto**

**Random copolymer adsorption**

The Morita approximation is a partial annealing procedure which yields upper bounds on the quenched average free energy. We consider higher order Morita approximations in which we control correlations to various orders between neighbouring monomers along the polymer chain. We consider different approaches for incorporating correlations and apply these to Motzkin and Dyck path models of the adsorption of a random copolymer at a surface. We also present lower bounds which, along with the Morita bounds, determine the limiting quenched average free energy for adsorption very precisely at low temperatures. Finally, Transfer Matrix methods are used to estimate the order and location of the phase transition.

**M. Biskup - University of California at Los Angeles
**Joint work (still in progress) with W. Koenig.

**Strong path localization for random walk in a random double-exponential potential**

I will describe recent progress in understanding path localization for a continuous time random walk in a random potential V. The model is as follows: A continuous random walk path of time-length T interacts with a random i.i.d. potential by picking energy proportional to the value of the potential at each site multiplied by the time it spends at that site. Using Feynman-Kac formula, the problem can be recast in terms of a parabolic Anderson model. This in turn permits the use of spectral analysis.

The main result is that, for a doubly exponential upper tail of the potential distribution, a typical path in a typical environment will dash towards a particular location in distance of order T log(T)logloglog(T) characterized by a favorable local eigenvalue of the Schroedinger operator Delta+V, and then stay in a vicinity of that point for the rest of the time. The new hard piece of information is that the location where the path "collapses" is uniquely determined for each realization of the environment. (Previously, the path was only known to pick one of T^{o(1)} such locations.) The analysis is based on deriving eigenvalue order statistics for the corresponding Schroedinger operator Delta+V.

**E. Bolthausen - Universität Zürich**

**On the localization-delocalization transition in a random copolymer model**

We discuss the critical localization-delocalization line for a copolymer with pinning at a one-dimensional defect line with random impurities. We prove that in a certain limit of strongly diluted strong disorder, the annealed critical point is different from the quenched one, the latter being in agreement with non-rigourous renormalization theory.

**F. Caravenna - Universitŕ degli Studi di Padova
**Joint work with G. Giacomin and M.
Gubinelli

**On the phase diagram of random copolymers at selective interfaces**

We consider a model of a random copolymer at a
selective interface which undergoes a localization/delocalization phase
transition. In spite of the several rigorous results available for this model,
there is still no agreement about some important issues. In particular, explicit
upper and lower bounds on the critical line have been proven, but these bounds
do not coincide.

In this talk we present numerical computations and observations, together with
rigorous arguments, which suggest that the critical line of the model lies
strictly in between the two bounds. In particular, we can perform a rigorous
statistical test showing that, with a very low level of error, the critical line
does not coincide with the lower bound. Finally, we show that the standard
implementation of the constrained annealing technique (also called Morita
approximation) cannot improve the upper bound on the critical line.

Presentation:

http://www.math.unipd.it/~fcaraven/download/slides/eindhoven_22-06-2007.pdf

**F. Comets -Université Paris 7
**Joint work with Serguei Popov (U. Sao Paulo) and Marina Vachkovskaia (Unicamp)

**On **
**-percolation
and the number of polymer paths**

Let be a sequence of independent Bernoulli variables, with parameter . If , the vertex is called open. Let be the number of oriented path of length which have exactly open vertices, and the number of those which have at least open vertices (). Note that are random variables. We prove that, for almost every ,

for all except maybe two values. The function is deterministic and relates to the free energy of directed polymers in random environment. This function has an explicit expression for and not too far from . Moreover, we obtain an equivalent for for and close to .

Presentation:

http://www.maths.dur.ac.uk/events/Meetings/LMS/2007/RW/Talks/comets.pdf

**J.D. Deuschel -Technische Universität Berlin
**Joint work with F. Caravenna

**Laplacian pinning and wetting
models in (1+1)-dimensions **

We consider a random line with a Laplacian interaction and delta-pinning of strength at the -axis (pinning model) respectively both pinning and repulsion above the -axis (wetting model). Unlike gradient systems with horizontally flat ground states, Laplacian models favor affine configurations and penalize curvature and bending. This model is used in the context of semi-flexible polymers and membranes. We show that this model undergoes pinning and wetting transitions: there are critical pinning, respectively wetting, parameters , such that the interface is delocalized for weak pinning constant and localized for strong pinning . We also derive a scaling limit theorem for the pinning model in the delocalized regime () and at criticality (). The proof is based on Markov renewal theory.

**
J. Dubbeldam - University
of Technology Delft**

**
Translocation of polymers through a narrow pore**

Translocation dynamics of a polymer chain through a nanopore has recently
attracted a lot of attention. The translocation process is known to play a
crucial role in the infection of cells by viruses and is also relevant to the
separation of DNA. We study the *dynamics* of polymer translocation by
scaling arguments, fractional calculus, and Monte Carlo simulations. To this end
the system is mapped on an one-dimensional *anomalous diffusion* process,
which is characterized by the number of translocated polymer segments
.
We find that the translocation time depends
on the polymer chain length through
a universal exponent whose
value is practically the same in two and three dimensions. The scaling and
analytical results for are
shown to be in good agreement with full three-dimensional Monte Carlo
simulations. Besides examing the translocation time, equations for the first and
second moments of will
be derived. The obtained expressions demonstrate once more that *polymer
translocation* is *anomalous* in nature.

The fact that diffusion through nanopores is anomalous in nature resolves an existing paradox in the literature that the time it takes for a polymer to diffuse through a pore is shorter than the time it takes to diffuse in absence of a wall with a pore.

**G. Giacomin - Université Paris 7**

**Pinning of polymers and renewal theory estimates**

**T. Guttmann -University of Melbourne
**Joint work with John Dethridge and Iwan Jensen

**Prudent self-avoiding walks and polygons. **

A recently proposed variant of self-avoiding walks (SAW) has been introduced in an unpublished manuscript by Pr\'ea, and more fully discussed by Duchi. Prudent self-avoiding walks (PSAW) are a proper subset of SAW, defined descriptively as follows: A walker, traversing a SAW, and looking straight ahead, cannot see any previously visited sites (no matter how far she looks ahead). For example, the sequence E,N,N,W,S is acceptable as a SAW, but is invalid as a PSAW. Prudent polygons are defined in terms PSAWS that terminate at a site adjacent to their starting point. In this talk I will discuss results obtained for both two-dimensional PSAWS and prudent polygons on the square lattice.

by the condition that the linear extension of any step, to arbitrary len

**R. van der Hofstad - Eindhoven University of Technology
**Joint work with Markus Heydenreich and Akira
Sakai.

**Mean-field behaviour for long-range
self-avoiding walks, percolation and the Ising model**

We consider self-avoiding walk, percolation and
the Ising model with long and finite range connections. Using the lace
expansion, we prove mean-field behavior for these models if $d>2(\alpha\wedge2)$
for self-avoiding walk and the Ising model, and $d>3(\alpha\wedge2)$ for
percolation, where $d$ denotes dimension and $\alpha$ the power-law decay
exponent of the step distribution.

We provide a simplified analysis based on trigonometry, which allows for
simplified proofs in the case where the coupling function has long-range
connections with power-law decay.

**G. Iliev - University of Toronto**

**Bicoloured Motzkin paths: A model of
unzipping duplex DNA**

Atomic force microscopy (AFM) and optical tweezer techniques allow individual polymer molecules to be micromanipulated. For instance, an adsorbed polymer can be pulled off a surface. We consider simple, exactly solvable models of this effect. In addition, we consider models of random copolymers adsorbed at an impenetrable surface and investigate their response to an elongational force. This is related to a model of unzipping duplex DNA.

**S. Lück -
Ulm University
**Joint work in progress with Wolfgang Arendt, Michael Beil, Frank Fleischer,
Stéphanie Portet and Volker Schmidt.

**A stochastic model for the reorganization of keratin networks **

Keratin networks play a dominant role in protecting the integrity of biological cells when they are exposed to mechanical stress and determine the elasticity of the cytoplasm. Changes of network morphology adjust the elastic properties of the cell and have e.g. been linked to the migration of cancer cells and metastasis. Network reorganization processes are, however, not well understood and cannot be studied by current microscopy techniques. A mathematical model will be presented that has been designed to study the effects of different hypothetical reorganization scenarios on network morphology by means of computer simulation. The model belongs to the class of piecewise-deterministic Markov processes. Its state space is a hybrid type incorporating a geometric component describing the network as well as a function space modelling concentration fields of soluble keratin in the cytoplasm. Diffusion of the soluble keratin in the cytoplasm has been described by means of a partial differential equation with periodic boundary conditions. Existence and uniqueness of a solution have been ensured by functional analytical techniques from the theory of bilinear forms. The talk will focus on introducing the biological background and giving an overview of the mathematical modelling concept. Some first simulation results will be presented.

**M.A.J. Michels - Eindhoven University of Technology**

**Disorder and criticality in polymer-like failure**

Polymers fail under large deformation by a characteristic sequence of phases: after an initial (quasi-)elastic regime the stress-strain curve generally shows a yield peak, announcing a mechanical instability and the onset of plasticity; after yield a stress-softening and flow regime sets in, but ultimately the material strain-hardens again before final fracture. The height of the yield peak and the extent of softening are believed to be related to slow local-ordering processes, while the hardening is in part due to the entangled-network topology. The balance between post-yield softening and subsequent hardening is known to be key in determining the macroscopic mechanical response: brittle or ductile. To understand essential features of the mechanical behaviour around the critical yield point, we model the polymer as a random network of two types of elastic springs: breakable springs with a spread in breaking thresholds to represent the weak cohesive forces, and unbreakable springs to represent the entangled covalent network. We study the statistics of damage development under deformation towards and beyond the critical yield point against two generic theories for critical behaviour: random percolation theory and self-organised criticality (SOC). Detailed comparisons of damage-cluster statistics and damage-avalanche statistics with theoretical scaling laws reveal how the presence of such generic critical behaviour depends on the disorder strength in the distribution of breaking thresholds of the weak cohesive bonds; random-percolation theory and SOC thereby represent opposite ends of the disorder spectrum.

**E. Orlandini - Universitŕ
degli Studi di Padova**

**Modelling polymers stretched by a force**

In recent years, the mechanical properties of individual polymers and
filaments have thoroughly investigated experimentally, thanks to the rapid
development of micromanipulation techniques such as optical tweezers and atomic
force microscopy (AFM). Experiments such as the stretching of single DNA
polymers or the force-induced desorption from an attractive surface enhance the
possibility of understanding the physical properties of the single molecule. In
order to interpret experiments quantitatively several theoretical models which
allow to calculate the response of a polymer to external forces have been
recently introduced and studied by

several authors. In this talk I will review some of these models showing that,
although they are simple enough to be solved
analytically, they can catch much of the underlying physics of the problem.
Moreover they can be extended to describe other
interesting phenomena such as the mechanical unzipping of double-stranded DNA.

**A. Owczarek - University of Melbourne**

**Polymers in a slab with attractive walls: Scaling and numerical results**

We analyse exact enumeration data and Monte Carlo simulation results for a self-avoiding walk model of a polymer confined between two parallel attractive walls (plates). We use the exact enumeration data to establish the regions where the polymer exerts an effective attractive force between the plates and where the polymer exerts an effective repulsive force by estimating the boundary (zero-force) curve. While the phase boundaries of the phase diagram have previously been conjectured we delineate this further by establishing the order of the phase transitions for the so-called infinite slab (that is, when the plates are a macroscopic distance apart). We conclude that the adsorption transitions associated with either plate are similar in nature to the half-space situation even when a polymer is attached to the opposite wall. The transition between the two adsorbed phases is established as first order. Importantly, we conjecture a scaling theory valid in the desorbed and critically adsorbed regions of the phase diagram and demonstrate the consistency of the Monte Carlo data with these hypotheses by estimating the corresponding scaling functions.

**N. Pétrélis -EURANDOM**

**Copolymer in a multi interface medium.
**

In this talk I will present some results that we obtained with Frank den Hollander about a model of copolymer in an medium made of droplets of a solvent of type $A$ in a medium of type $B$. This model was introduced by F. den Hollander and S. Whittington in their paper Diffusion of a heteropolymer in a multi interface medium. We will focus on the super-critical case (when the droplets of type $A$ percolates), and more particularly on the phase transition between full-delocalization in the infinite cluster of $A$ and partial-localization at the interface between the infinite cluster and the solvent $B$. We will see that the order of the transition is exactly 2.

**A. Rechnitzer - University of British Columbia**

**Combinatorial models of polymers in a slit
with attractive walls**

Directed paths on regular lattices are idealised geometric models of polymers. Despite their apparent simplicity, they give rise to interesting combinatorial problems and display rich behaviour. In this talk I will discuss a directed walk model of polymer-colloid interactions. The generating functions for this model (and a number of generalisations of it) can be computed exactly using a variety of techniques.From these, we are able to extract information about the behaviour of the model in the long-polymer limit and demonstrate that the model exhibits several different thermodynamic phases.

**E.J. Janse van Rensburg - York University**

**Directed paths in a wedges**

The enumeration of lattice paths remains an interesting and productive area
in statistical mechanics and in combinatorics. In statistical mechanics, models
of lattice paths arise naturally as models of polymers, and directed versions of
these models pose challenging combinatorial questions, some remaining beyond
current mathematical techniques. In this talk I review the enumeration of
directed and partially directed lattice paths confined to wedges using an
iterated version of the kernel method. I shall show how to determine the
generating function for the number of paths in two simple models and discuss
asymptotic formulae for the number of paths.

This work was done in collaboration with Andrew Rechnitzer and Thomas Prellberg.

**C. Richard -
Universität Bielefeld**

**Some problems in the statistical mechanics of polymers **

Analysing thermodynamic quantities related to polymer models is a probabilistic problem. Away from phase transition points, thermodynamic quantities typically only show microscopic fluctuations, resulting in Gaussian limit laws. In contrast, at phase transition points, macroscopic fluctuations are expected, leading to non-Gaussian limit laws.

We will give an overview of rigorous results for these questions, compare techniques, mainly from statistical mechanics and asymptotic combinatorics, and state some open questions.

**V. Sidoravicius - IMPA**

**Columnar effects and pinning for a few oriented polymer models
**

We present new results on the effect of a "columnar defect" (i.e. of a perturbation of the environment along a line) on various models related to directed polymers driven by a random two-dimensional Poisson process.

Related to these are the so-called "slow-bond problem" for the asymmetric exclusion process, and a new process defined in terms of interacting and annihilating random walks. The main tool used in the study of these problems, even though the specific details differ in each case, is that of "influence percolation", i.e. a way to relate these problems to one of one-dimensional, dependent, long-range percolation.

**C. Soteros - University of Saskatchewan**

**Bounds for directed walk models of random copolymer localization**

The focus will be on bounds for the limiting quenched average free energy for bilateral Dyck and Motzkin path models of random copolymer localization. Upper bounds are obtained through a sequence of higher order Morita approximations and lower bounds come from finite size quenched average free energies (exact enumeration) and their lower bounds.

**G. Slade, University of British Columbia**

**The lace expansion and the enumeration of self-avoiding walks**

The lace expansion is an elegant combinatorial construction that provides a recursion relation for the number of self-avoiding walks.

We first give an introduction to the lace expansion, and then explain how it has been used recently (in joint work with Nathan Clisby and Richard Liang) to enumerate self-avoiding walks on the hypercubic lattice up to n=30 steps in dimension 3, and up to n=24 steps in all dimensions above 3. Major improvements to the 1/d expansion for the connective constant have also been obtained. In addition, an algorithmic improvement called the two-step method will be described.

**F.L. Toninelli - Laboratoire de Physique, ENS Lyon**

**A replica-coupling approach to disordered pinning models**

I will consider models of directed polymers pinned by 1-dimensional quenched
randomness. These models undergo a localization/delocalization transition. I
will focus on the effect of disorder on the phase transition. Disorder is
expected to be relevant (i.e., to change the order of the transition) or
irrelevant according to whether a certain critical exponent of the homogeneous (non-disordered)
model is positive or not.

This can be proven rigorously. In my talk, I will concentrate on the "irrelevance"
question, which was first solved by K. Alexander. I will show how a
generalization of "interpolation" and "replica coupling" techniques, well known
in spin glass, allows to re-obtain this result in a simple way and to get
sharper estimates on the free energy close to criticality.

*This page is maintained by **Lucienne
Coolen*