F. Bailly

CONFIGURATIONAL ENTROPY OF CODIMENSION-ONE TILINGS AND DIRECTED MEMBRANES

The calculation of random tiling configurational entropy amounts to an enumeration of partitions. A geometrical description of the configuration space is given in terms of integral points in a high-dimensional space, and the entropy is deduced from the integral volume of a convex polytope. In some cases the latter volume can be expressed in a compact multiplicative formula, and in all cases in terms of binomial series, the origin of which is given a geometrical meaning. Our results mainly concern codimension-one tilings, but can also be extended to higher codimension tilings. We also discuss the link between freeboundary-and fixed-boundary-condition problems.

Fixed-Boundary Octagonal Random Tilings: A Combinatorial Approach

Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus generalizing previous results on the more restricted class of codimension-one tilings. In particular this method gives access to counting formulas, which are directly related to questions of entropy in these statistical systems. Methods and tools from the field of enumerative combinatorics are used.

Arctic Octahedron in Three-Dimensional Rhombus Tilings and Related Integer Solid Partitions

Three-dimensional integer partitions provide a convenient representation of codimension-one three-dimensional random rhombus tilings. Calculating the entropy for such a model is a notoriously difficult problem. We apply transition matrix Monte Carlo simulations to evaluate their entropy with high precision. We consider both free- and fixed-boundary tilings. Our results suggest that the ratio of free- and fixed-boundary entropies is σfree/σfixed=3/2, and can be interpreted as the ratio of the volumes of two simple, nested, polyhedra. This finding supports a conjecture by Linde, Moore, and Nordahl concerning the “arctic octahedron phenomenon” in three-dimensional random tilings.

Configurational entropy in random tiling models

Abstract The configurational entropy of some random tiling models is calculated. We mainly focus on co-dimension one rhombohedral tilings for which a general approximate formula is given for the number of configurations as a function of boundary conditions. The formula gives numbers remarkably close to (numerically derived) exact values and is believed to be valid in the thermodynamic limit. The link with a related statistical mechanics problem is pointed out.

Random tilings of high symmetry: I. Mean-field theory

We study random tiling models in the limit of high rotational symmetry. In this limit a mean-field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions. We base our mean-field theory on an iterative tiling construction inspired by the work of de Bruijn. In addition to the entropy, we consider correlation functions, phason elasticity and the thermodynamic limit. Tilings of dimension other than two are considered briefly.

Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies

We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in the limit of continuous symmetry. Tilings with phason strain appear to share the same entropy as unstrained tilings, as predicted by mean-field theory. We consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy, although these differ for finite rotational symmetry.

A formula for the number of tilings of an octagon by rhombi

We propose the first algebraic determinantal formula to enumerate tilings of a centrosymmetric octagon of any size by rhombi. This result uses the Gessel-Viennot technique and generalizes to any octagon a formula given by Elnitsky in a special case.

Two-dimensional random tilings of large codimension: new progress

Two-dimensional random tilings of rhombi can be seen as projections of two-dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a D-dimensional space. We study the limiting case, when the quantity D, and therefore the number of different species of tiles, become large. We had previously demonstrated [M. Widom, N. Destainville, R. Mosseri, F. Bailly, in: Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1997.] that, in this limit, the thermodynamic properties of the tiling become independent of the boundary conditions. The exact value of the limiting entropy and finite D corrections remain open questions. Here, we develop a mean-field theory, which uses an iterative description of the tilings based on an analogy with avoiding oriented walks on a random tiling. We compare the quantities so-obtained with numerical calculations. We also discuss the role of spatial correlations.

Anti-E1E2 antibodies do predict response to triple therapy in treatment-experienced Hepatitis C Virus-cirrhosis cases.

BACKGROUND AND AIMSnWe previously showed that pre-treatment serum anti-E1E2 predicted hepatitis C virus (HCV) RNA viral kinetics (VKs) and treatment outcome in patients with chronic hepatitis C receiving pegylated interferon/ribavirin (Peg-IFN/RBV) double therapy. Here, we determined whether baseline anti-E1E2 was correlated with the on-treatment VK and could predict virological outcome in treatment-experienced HCV-infected cirrhotic patients receiving protease inhibitor-based triple therapy.nnnMETHODSnSera from 19 patients with HCV genotype 1 infection and compensated cirrhosis who failed to respond to a prior course of Peg-IFN/RBV were selected at time 0 before starting triple therapy with boceprevir or telaprevir. We assessed patients with sustained viral response 12 weeks after the end of triple therapy (SVR12) by analyzing VKs at weeks 4, 12, 24, 36, 48 (end of treatment) and 60.nnnRESULTSnPatients baseline characteristics were similar to the well-defined CUPIC cohort (age, HCV subtype, baseline viremia, and treatment history). Among the 19 patients, 11 achieved an SVR12. Fifteen patients were positive for pre-treatment anti-E1E2 and all of them achieved SVR12. Moreover, anti-E1E2 and SVR12 correlated with prior response to IFN/RBV therapy (relapse, partial or null response).nnnCONCLUSIONSnBaseline anti-E1E2 could be considered as a new biomarker to predict SVR12 after triple therapy in this most difficult-to-treat population. These results warrant further validation on larger cohorts including patients receiving highly effective direct-acting antivirals to explore whether this test could help in better defining treatment duration for these very costly molecules.