Nicolas Pétrélis

A Polymer in a Multi-Interface Medium

We consider a model for a polymer chain interacting with a sequence of equispaced flat interfaces through a pinning potential. The intensity d?R of the pinning interaction is constant, while the interface spacing T=TN is allowed to vary with the size N of the polymer. Our main result is the explicit determination of the scaling behavior of the model in the large N limit, as a function of (TN)N and for fixed d>0. In particular, we show that a transition occurs at TN=O(log?N). Our approach is based on renewal theory.

A Variational Formula for the Free Energy of the Partially Directed Polymer Collapse

Long linear polymers in dilute solutions are known to undergo a collapse transition from a random coil (expand itself) to a compact ball (fold itself up) when the temperature is lowered, or the solvent quality deteriorates. A natural model for this phenomenon is a 1+1 dimensional self-interacting and partially directed self-avoiding walk. In this paper, we develop a new method to study the partition function of this model, from which we derive a variational formula for the free energy. This variational formula allows us to prove the existence of the collapse transition and to identify the critical temperature in a simple way. We also provide a probabilistic proof of the fact that the collapse transition is of second order with critical exponent 3/2.

On the Localized Phase of a Copolymer in an Emulsion: Subcritical Percolation Regime

The present paper is a continuation of the authors work “EURANDOM Report 2007-048”. The object of interest is a two-dimensional model of a directed copolymer, consisting of a random concatenation of hydrophobic and hydrophilic monomers, immersed in an emulsion, consisting of large blocks of oil and water arranged in a percolation-type fashion. The copolymer interacts with the emulsion through an interaction Hamiltonian that favors matches and disfavors mismatches between the monomers and the solvents, in such a way that the interaction with the oil is stronger than with the water.The model has two regimes, supercritical and subcritical, depending on whether the oil blocks percolate or not. In our work “EURANDOM Report 2007-048” we focussed on the supercritical regime and obtained a complete description of the phase diagram, which consists of two phases separated by a single critical curve. In the present paper we focus on the subcritical regime and show that the phase diagram consists of four phases separated by three critical curves meeting in two tricritical points.

Interacting partially directed self avoiding walk: scaling limits

This paper is dedicated to the investigation of a

Scaling limit of the uniform prudent walk

1+1

Annealed scaling for a charged polymer

dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW and introduced in \cite{ZL68} by Zwanzig and Lauritzen to study the collapse transition of an homopolymer dipped in a poor solvant. In \cite{POBG93}, physicists displayed numerical results concerning the typical growth rate of some geometric features of the path as its length

Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase

L

The discrete-time parabolic Anderson model with heavy-tailed potential

diverges. From this perspective the quantities of interest are the projections of the path onto the horizontal axis (also called horizontal extension) and onto the vertical axis for which it is useful to define the lower and the upper envelopes of the path. With the help of a new random walk representation, we proved in \cite{CNGP13} that the path grows horizontally like

Lectures on random polymers

\sqrt{L}

A mathematical model for a copolymer in an emulsion

in its collapsed regime and that, once rescaled by