1. The random pinning model with correlated disorder given by a renewal set
- Author
-
Dimitris Cheliotis, Julien Poisat, Yuki Chino, Department of Mathematics [Athens], National and Kapodistrian University of Athens (NKUA), Mathematisch Instituut Universiteit Leiden, Mathematical institute, Universiteit Leiden [Leiden]-Universiteit Leiden [Leiden], CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and CNRS PEPS
- Subjects
Physics ,Pure mathematics ,Phase transition ,010102 general mathematics ,Probability (math.PR) ,Second moment of area ,01 natural sciences ,Critical point (mathematics) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Exponent ,Decoupling (probability) ,FOS: Mathematics ,0101 mathematics ,Completeness (statistics) ,Smoothing ,Mathematics - Probability - Abstract
We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha$ > 0, when the correlated sequence is given by another independent renewal set with loop exponent $\alpha$ > 0. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $\alpha$ > 2 (summable correlations), disorder is irrelevant if $\alpha$ < 1/2 and relevant if $\alpha$ > 1/2, which extends the Harris criterion for independent disorder. The case $\alpha$ $\in$ (1, 2) (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha$ > 1/ $\alpha$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\alpha$ $\in$ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.
- Published
- 2017