Estimates on path delocalization for copolymers at selective interfaces

• Published in: Probability theory and related fields Vol. 133; no. 4; pp. 464 - 482
• Main Authors: GIACOMIN, Giambattista, TONINELLI, Fabio Lucio
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.12.2005; Berlin Springer Nature B.V; New York, NY Springer Verlag
• Subjects: Condensed Matter; Continuity (mathematics); Copolymers; Exact sciences and technology; Fluctuation phenomena, random processes, noise, and brownian motion; Free energy; Inequality; Interfaces; Interpolation; Lattice theory and statistics (ising, potts, etc.); Localization; Mathematics; Partitions (mathematics); Physics; Polymers; Probability; Probability and statistics; Probability theory and stochastic processes; Random variables; Random walk; Random walks and levy flights; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistical Mechanics; Statistical physics, thermodynamics, and nonlinear dynamical systems; Studies; 60K35, 82B41, 82B44 Copolymers; Random walk; Delocalization transition; Polymer; Continuous function; Probability distribution; Interpolation techniques; Free energy; Concentration inequalities; Interpolation; Partition function; Increasing function; Directed Polymers; Random variable; Copolymers; Delocalization Transition
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-005-0439-2

Starting from the simple symmetric random walk {Sn}n, we introduce a new process. We are looking for results in the large N limit. This factor favors Sn>0 if Wn+h>0 and Sn<0 if Wn+h<0. The process can be interpreted as a model for a random heterogeneous polymer in the proximity of an interface separating two selective solvents. It has been shown [6] that this model undergoes a (de)localization transition: more precisely there exists a continuous increasing function ??hc(?) such that if hhc(?). In particular we prove that, in a suitable sense, one cannot expect more than O( log N) visits of the walk to the lower half plane. The previously known bound was o(N). Stronger O(1)-type results are obtained deep inside the delocalized region. The same approach is also helpful for a different type of question: we prove in fact that the limit as ? tends to zero of hc(?)/? exists and it is independent of the law of ?1, at least when the random variable ?1 is bounded or it is Gaussian. This is achieved by interpolating between this class of variables and the particular case of ?1 taking values ±1 with probability 1/2, treated in [6]. [PUBLICATION ABSTRACT]