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

In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p \in (0,1)$. The copolymer in the emulsion has an energy that is minus $\alpha$ times the number of $AA$-matches minus $\beta$ times the number of $BB$-matches, where without loss of generality the interaction parameters can be taken from the cone $\{(\alpha,\beta)\in\R^2\colon \alpha\geq |\beta|\}$. To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In \cite{dHW06}, it was found that in the supercritical percolation regime $p \geq p_c$, with $p_c$ the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of $p$. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the $A$-blocks to a phase where it is partially localized near the $AB$-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase.Comment: 43 pages and 10 figure

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

Article / Letter to editorMathematisch Instituu

A mathematical model for a copolymer in an emulsion

Analysis and Stochastic

Phase diagram for a copolymer in a micro-emulsion

Analysis and Stochastic

Properties of additive functionals of Brownian motion with resetting

We study the distribution of additive functionals of reset Brownian motion, a variation of normal Brownian motion in which the path is interrupted at a given rate and placed back to a given reset position. Our goal is two-fold: (1) For general functionals, we derive a large deviation principle in the presence of resetting and identify the large deviation rate function in terms of a variational formula involving large deviation rate functions without resetting. (2) For three examples of functionals (positive occupation time, area and absolute area), we investigate the effect of resetting by computing distributions and moments, using a formula that links the generating function with resetting to the generating function without resetting.Comment: 23 pages, 4 figure

Missense mutations at homologous positions in the fourth and fifth laminin A G-like domains of eyes shut homolog cause autosomal recessive retinitis pigmentosa

Purpose: To describe two novel mutations in the eyes shut homolog (EYS) gene in two families with autosomal recessive retinitis pigmentosa (arRP) from Pakistan and Indonesia. Methods: Genome-wide linkage and homozygosity mapping were performed using single nucleotide polymorphism microarray analysis in affected members of the two arRP families. Sequence analysis was performed to identify genetic changes in protein coding exons of EYS. Results: In the Indonesian and Pakistani families, homozygous regions encompassing the EYS gene at 6q12 were identified, with maximum LOD scores of 1.8 and 3.6, respectively. Novel missense variants in the EYS gene (p.D2767Y and p.D3028Y) were found in the Pakistani and Indonesian families, respectively, that co-segregate with the disease phenotype. Interestingly, the missense variants are located at the same homologous position within the fourth and fifth laminin A G-like domains of EYS. Conclusions: To date, mostly protein-truncating mutations have been described in EYS, while only few patients have been described with pathogenic compound heterozygous missense mutations. The mutations p.D2767Y and p.D3028Y described in this study affect highly conserved residues at homologous positions in laminin A G-like domains and support the notion that missense mutations in EYS can cause arRP

Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States

The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We introduce the notion of relaxation height in a general energy landscape and we prove sufficient conditions which are valid even in presence of multiple metastable states. We show how these results can be used to approach the problem of multiple metastable states via the use of the modern theories of metastability. We finally apply these general results to the Blume--Capel model for a particular choice of the parameters ensuring the existence of two multiple, and not degenerate in energy, metastable states