Annealed scaling for a charged polymer in dimensions two and higher

作者：Berger Q^*; den Hollander F; Poisat J

来源：Journal of Physics A-Mathematical and Theoretical, 2018, 51(5): 054002.

DOI：10.1088/1751-8121/aa9f83

• charged polymer
• annealed free energy
• phase transition
• collapsed phase
• extended phase
• scaling
• large deviations

This paper considers an undirected polymer chain on Z(d), d >= 2, with i.i.d. random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of die 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 object of interest is the annealed free energy per monomer in the limit as the length n of the polymer chain tends to infinity.
We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an extended phase from a collapsed phase. We derive the scaling of the critical curve for small and for large charge bias and the scaling of the annealed free energy for small inverse temperature. We argue that in the collapsed phase the polymer chain is subdiffusive, namely, on scale (n/log n)(1/(d+2)) it moves like a Brownian motion conditioned to stay inside a ball with a deterministic radius and a randomly shifted center. We further expect that in the extended phase the polymer chain scales like a weakly self-avoiding walk.
The scaling of the critical curve for small charge bias and the scaling of die annealed free energy for small inverse temperature arc both anomalous. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the self-intersection local time and the upward large deviations of the range. Part of our scaling results are rough. We formulate conjectures under which they can be sharpened. The existence of the free energy remains an open problem, which we are able to settle in a subset of the collapsed phase for a subclass of charge distributions.

• 出版日期2018-2-2