arXiv subjects

Nicolas Petrelis

Publications and source records attributed to Nicolas Petrelis.

3 recordsLinked to original sources

arXiv2012

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 prove that the order of the collapse transition is 3/2.

math.PR↗

arXiv2008

On the localized phase of a copolymer in an emulsion: subcritical percolation regime

The present paper is a continuation of \cite{dHP07b}. 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 \cite{dHP07b} 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.

math.PR↗

arXiv2007

Copolymer at selective interfaces and pinning potentials: weak coupling limits

We consider a simple random walk of length $N$, denoted by $(S_{i})_{i\in \{1,...,N\}}$, and we define $(w_i)_{i\geq 1}$ a sequence of centered i.i.d. random variables. For $K\in\N$ we define $((γ_i^{-K},...,γ_i^K))_{i\geq 1}$ an i.i.d sequence of random vectors. We set $β\in \mathbb{R}$, $λ\geq 0$ and $h\geq 0$, and transform the measure on the set of random walk trajectories with the Hamiltonian $λ\sum_{i=1}^{N} (w_i+h) \sign(S_i)+β\sum_{j=-K}^{K}\sum_{i=1}^{N} γ_{i}^{j} \boldsymbol{1}_{\{S_{i}=j\}}$. This transformed path measure describes an hydrophobic(philic) copolymer interacting with a layer of width $2K$ around an interface between oil and water. In the present article we prove the convergence in the limit of weak coupling (when $λ$, $h$ and $β$ tend to 0) of this discrete model towards its continuous counterpart. To that aim we further develop a technique of coarse graining introduced by Bolthausen and den Hollander in \cite{BDH}. Our result shows, in particular, that the randomness of the pinning around the interface vanishes as the coupling becomes weaker.

math.PR↗