Annealed scaling for a charged polymer in dimensions two and higher

Abstract

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 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 show that in a subset of 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 expect this scaling to hold throughout the collapsed phase. 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 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.

Transience and Recurrence of Markov Processes with Constrained Local Time

“…The goal is often to understand when the transition occurs as underlying model parameters vary, as in e.g. Bolthausen (2002); Berger et al (2018); Caravenna et al (2012); den Hollander (1999); Hu et al (2011). This motivates our study of transience and recurrence of Markov processeses, transience and recurrence corresponding to delocalised and localised phases, respectively.…”

“…The (right-continuous) inverse of the local time process is a subordinator (a non-decreasing Lévy process), so our study of the behaviour of the local time process is effectively equivalent to studying a subordinator conditioned to grow faster than the inverse function f −1 . Berger et al (2018); Caravenna et al (2012); van der Hofstad and König (2001); van der Hofstad and Klenke (2001); Westwater (1980). Particularly important is the transition between a localised phase, where the polymer remains close to a point, and a delocalised phase, where it moves away.…”

Annealed Scaling for a Charged Polymer

“…The terms arising from G (5) in (C.64) can be shown to be of negligible order. Therefore we obtain, for β small enough, Proof of (iii) in (C.3).…”

“…The Ray-Knight formula is no longer available in d ≥ 2. In Berger, den Hollander and Poisat [5] it is shown that the phase diagram is qualitatively similar, but no detailed description of the scaling behaviour in the two phases is obtained.…”

“…We expand e e δ,β ( ) − 1 to fifth order in β, so that the expansion includes the limiting term. There exists a constant c such that (4.78) e e δ,β ( ) − 1 ≤ e δ,β ( ) + 1 2 e δ,β ( ) 2 + 1 6 e δ,β ( ) 3 + 1 24 e δ,β ( ) 4 + ce δ,β ( ) 5…”