The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics

We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$ on Z^d with jump rate rho (that plays the role of the random medium), we modify the law of a random walk X on Z^d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t \ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta L_t(X,Y)), beta in R. As beta increases, the system exhibits a delocalization/localization transition: there is a critical value beta_c, such that if beta>beta_c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d larger or equal to three 3, the presence of disorder makes the phase transition at least of second order. This, in dimension larger or equal to 4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.Comment: 18 page

New bounds for the free energy of directed polymers in dimension 1+1 and 1+2

We study the free energy of the directed polymer in random environment in dimension 1+1 and 1+2. For dimension 1, we improve the statement of Comets and Vargas concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension 2, we prove that very strong disorder holds at all temperatures, thus solving a long standing conjecture in the field.Comment: 31 pages, 4 figures, final version, accepted for publication in Communications in Mathematical Physic

Stretched Polymers in Random Environment

We survey recent results and open questions on the ballistic phase of stretched polymers in both annealed and quenched random environments.Comment: Dedicated to Erwin Bolthausen on the occasion of his 65th birthda

Variational formulas and cocycle solutions for directed polymer and percolation models

We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder

Gastrointestinal symptoms and association with medication use patterns, adherence, treatment satisfaction, quality of life, and resource use in osteoporosis: baseline results of the MUSIC-OS study

Summary: The Medication Use Patterns, Treatment Satisfaction, and Inadequate Control of Osteoporosis Study (MUSIC-OS) is a prospective, observational study of women with osteoporosis in Europe and Canada. At baseline, patients with gastrointestinal symptoms reported lower adherence to osteoporosis treatment, treatment satisfaction, and health-related quality of life, than those without gastrointestinal symptoms. Introduction: The aim of the study was to examine gastrointestinal (GI) symptoms and the association between GI symptoms and treatment adherence, treatment satisfaction, and health-related quality of life (HRQoL) among osteoporotic women in Europe and Canada. Methods: Baseline results are reported here for a prospective study which enrolled postmenopausal, osteoporotic women who were initiating (new users) or continuing (experienced users) osteoporosis treatment at study entry (baseline). A patient survey was administered at baseline and included the occurrence of GI symptoms during 6-month pre-enrolment, treatment adherence (adherence evaluation of osteoporosis (ADEOS), score 0–22), treatment satisfaction (Osteoporosis Treatment Satisfaction Questionnaire for Medications (OPSAT-Q), score 0–100) and HRQoL (EuroQol-5 dimension (EQ-5D) utility, score 0–1; OPAQ-SV, score 0–100). The association between GI symptoms and ADEOS (experienced users), OPSAT-Q (experienced users), and HRQoL (new and experienced users) was assessed by general linear models adjusted for patient characteristics. Results: A total of 2959 patients (2275 experienced and 684 new users) were included. Overall, 68.1 % of patients experienced GI symptoms in the past 6 months. Compared with patients without GI symptoms, patients with GI symptoms had lower mean baseline scores on most measures. The mean adjusted differences were ADEOS, −0.43; OPSAT-Q, −5.68; EQ-5D, −0.04 (new users) and −0.06 (experienced users), all P < 0.01. GI symptoms were also associated with lower OPAQ-SV domain scores: physical function, −4.17 (experienced users); emotional status, −4.28 (new users) and −5.68 (experienced users); back pain, −5.82 (new users) and −11.33 (experienced users), all P < 0.01. Conclusions: Patients with GI symptoms have lower treatment adherence and treatment satisfaction and worse HRQoL than patients without GI symptoms