Polymer dynamics in the depinned phase: metastability with logarithmic barriers

• Published in: Probability theory and related fields Vol. 153; no. 3-4; pp. 587 - 641
• Main Authors: Caputo, Pietro, Lacoin, Hubert, Martinelli, Fabio, Simenhaus, François, Toninelli, Fabio Lucio
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-08-2012; Springer Nature B.V; Springer Verlag
• Subjects: Asymptotic properties; Economics; Energy; Equilibrium; Evolution; Finance; Insurance; Localization; Management; Markov analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Phase transitions; Polymers; Polynomials; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Statistics for Business; Theoretical; Thermalization (energy absorption); Reversible Markov chains; Polymer pinning model; Spectral gap; Mixing time; Quasi-stationary distribution; Coupling; 82C20; 60K35; Metastability
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-011-0355-6

We consider the stochastic evolution of a (1 + 1)-dimensional polymer in the depinned regime. At equilibrium the system exhibits a double well structure: the polymer lies (essentially) either above or below the repulsive line. As a consequence, one expects a metastable behavior with rare jumps between the two phases combined with a fast thermalization inside each phase. However, the energy barrier between these two phases is only logarithmic in the system size
L
and therefore the two relevant time scales are only polynomial in
L
with no clear-cut separation between them. The whole evolution is governed by a subtle competition between the diffusive behavior inside one phase and the jumps across the energy barriers. Our main results are: (i) a proof that the mixing time of the system lies between
and
; (ii) the identification of two regions associated with the positive and negative phase of the polymer together with the proof of the asymptotic exponentiality of the tunneling time between them with rate equal to a half of the spectral gap.