Annealed Scaling for a Charged Polymer

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• 作者：F. Caravenna ; F. den Hollander ; N. Pétrélis…
• 关键词：Charged polymer ; Quenched vs. annealed free energy ; Large deviations ; Phase transition ; Ballistic vs. subballistic phase ; Scaling ; 60K37 ; 82B41 ; 82B44
• 刊名：Mathematical Physics, Analysis and Geometry
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：19
• 期：1
• 全文大小：2,279 KB
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• 作者单位：F. Caravenna (1)
  F. den Hollander (2)
  N. Pétrélis (3)
  J. Poisat (4)
  
  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 55, 20125, Milano, Italy
  2. Mathematical Institute, Leiden University, P.O. Box 9512, 2300, RA, Leiden, The Netherlands
  3. Laboratoire de Mathématiques Jean Leray UMR 6629, Université de Nantes, 2 Rue de la Houssinière, BP 92208, 44322, Nantes Cedex 03, France
  4. CEREMADE, UMR 7534, Université Paris-Dauphine, PSL Research University, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Analysis
  Geometry
  Group Theory and Generalizations
  Applications of Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9656

文摘

This paper studies an undirected polymer chain living on the one-dimensional integer lattice and carrying i.i.d. random charges. Each self-intersection of the polymer chain contributes to the interaction Hamiltonian an energy that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The focus is on the annealed free energy per monomer in the limit as the length of the polymer chain tends to infinity. We derive a spectral representation for the free energy and use this to prove that there is a critical curve in the parameter plane of charge bias versus inverse temperature separating a ballistic phase from a subballistic phase. We show that the phase transition is first order. We prove large deviation principles for the laws of the empirical speed and the empirical charge, and derive a spectral representation for the associated rate functions. Interestingly, in both phases both rate functions exhibit flat pieces, which correspond to an inhomogeneous strategy for the polymer to realise a large deviation. The large deviation principles in turn lead to laws of large numbers and central limit theorems. We identify the scaling behaviour of the critical curve for small and for large charge bias. In addition, we identify the scaling behaviour of the free energy for small charge bias and small inverse temperature. Both are linked to an associated Sturm-Liouville eigenvalue problem. A key tool in our analysis is the Ray-Knight formula for the local times of the one-dimensional simple random walk. This formula is exploited to derive a closed form expression for the generating function of the annealed partition function, and for several related quantities. This expression in turn serves as the starting point for the derivation of the spectral representation for the free energy, and for the scaling theorems. What happens for the quenched free energy per monomer remains open. We state two modest results and raise a few questions. Keywords Charged polymer Quenched vs. annealed free energy Large deviations Phase transition Ballistic vs. subballistic phase Scaling