English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This paper considers an undirected polymer chain on $\mathbb{Z}^d$, $d \geq2$, with i.i.d.\ random charges attached to its constituent monomers. Eachself-intersection of the polymer chain contributes an energy to the interactionHamiltonian that is equal to the product of the charges of the two monomersthat meet. The joint probability distribution for the polymer chain and thecharges is given by the Gibbs distribution associated with the interactionHamiltonian. The object of interest is the \emph{annealed free energy} permonomer 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 thecharge bias and the inverse temperature separating an \emph{extended phase}from a \emph{collapsed phase}. We derive the scaling of the critical curve forsmall and for large charge bias and the scaling of the annealed free energy forsmall inverse temperature, which are both anomalous. We show that in a subsetof the collapsed phase the polymer chain is \emph{subdiffusive}, namely, onscale $(n/\log n)^{1/(d+2)}$ it moves like a Brownian motion conditioned tostay inside a ball with a deterministic radius and a randomly shifted center.We expect this scaling to hold throughout the collapsed phase. We furtherexpect that in the extended phase the polymer chain scales like a weaklyself-avoiding walk. Proofs are based on a detailed analysis for simple randomwalk of the downward large deviations of the self-intersection local time andthe upward large deviations of the range. Part of our scaling results arerough, and we formulate conjectures under which they can be sharpened. Theexistence of the free energy remains an open problem, which we are able tosettle in a subset of the collapsed phase for a subclass of chargedistributions.

Annealed scaling for a charged polymer in dimensions two and higher