Phase transitions for spatially extended pinning

Probability Theory and Related Fields, vol. 181, pp. 329–375

Abstract

Abstract We consider a directed polymer of lengthNinteracting with a linear interface. The monomers carry i.i.d. random charges $$(\omega _i)_{i=1}^N$$ (ωi)i=1N taking values in $${\mathbb {R}}$$ R with mean zero and variance one. Each monomericontributes an energy $$(\beta \omega _i-h)\varphi (S_i)$$ (βωi-h)φ(Si) to the interaction Hamiltonian, where $$S_i \in {\mathbb {Z}}$$ Si∈Z is the height of monomeriwith respect to the interface, $$\varphi :\,{\mathbb {Z}}\rightarrow [0,\infty )$$ φ:Z→[0,∞) is the interaction potential, $$\beta \in [0,\infty )$$ β∈[0,∞) is the inverse temperature, and $$h \in {\mathbb {R}}$$ h∈R is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on $${\mathbb {Z}}$$ Z . We study both thequenchedand theannealedfree energy per monomer in the limit as $$N\rightarrow \infty $$ N→∞ . We show that each exhibits a phase transition along a critical curve in the $$(\beta ,h)$$ (β,h) -plane, separating alocalized phase(where the polymer stays close to the interface) from adelocalized phase(where the polymer wanders away from the interface). We derive variational formulas for the critical curves and we obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by aBessel random walk, we derive the scaling limit of the annealed free energy as $$\beta ,h \downarrow 0$$ β,h↓0 inthree different regimesfor the tail exponent of $$\varphi $$ φ .

Authors 2

1. Francesco Caravenna

   University of Milano-Bicocca

   Affiliation as printed

   Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via Cozzi 55, 20125, Milan, Italy

   Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Milan, Italy#TAB#

2. Frank den Hollander corresponding Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch instituut, Universiteit Leiden, Postbus 9512, 2300, RA, Leiden, The Netherlands

   Mathematisch Instituut, Universiteit Leiden, Leiden, the Netherlands