Annealed scaling for a charged polymer in dimensions two and higher
Abstract
This paper considers an undirected polymer chain on {Z}^{d} , d ≥slant 2 , with i.i.d. random charges attached to its constituent monomers. Each selfintersection of the polymer chain contributes an energy to the interaction Hamiltonian 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 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 \newcommand{\hs}{\hat{s}} (n/log n){\hspace{0pt}}^{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 selfavoiding walk.
The scaling of the critical curve for small charge bias and the scaling of the annealed free energy for small inverse temperature are both anomalous. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the selfintersection 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.
The research in this paper was supported through ERC Advanced Grant 267356VARIS.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 February 2018
 DOI:
 10.1088/17518121/aa9f83
 arXiv:
 arXiv:1708.06707
 Bibcode:
 2018JPhA...51e4002B
 Keywords:

 Mathematical Physics;
 Mathematics  Probability;
 82D60;
 60K35;
 82B27
 EPrint:
 35 pages, 6 figures