Overlap properties and adsorption transition of two Hamiltonian paths

TL;DR

This paper models two compact polymer chains as Hamiltonian paths with attractive interactions, revealing a phase transition to a frozen state where one chain is fully adsorbed onto the other, and analyzes their overlap properties.

Contribution

It introduces a spin representation for coupled Hamiltonian paths and demonstrates a phase transition to a frozen, adsorbed phase with detailed overlap distribution analysis.

Findings

Existence of a phase transition at low temperature to a frozen adsorbed phase.

Bounds on the overlap between two Hamiltonian paths, with complete coincidence above a certain threshold.

Probability distribution of overlaps derived via Legendre transform.

Abstract

We consider a model of two (fully) compact polymer chains, coupled through an attractive interaction. These compact chains are represented by Hamiltonian paths (HP), and the coupling favors the existence of common bonds between the chains. Using a ($n=0$ component) spin representation for these paths, we show the existence of a phase transition for strong coupling (i.e. at low temperature) towards a ``frozen'' phase where one chain is completely adsorbed onto the other. By performing a Legendre transform, we obtain the probability distribution of overlaps. The fraction of common bonds between two HP, i.e. their overlap $q$, has both lower ($q_m$) and upper ($q_M$) bounds. This means in particuliar that two HP with overlap greater than $q_M$ coincide. These results may be of interest in (bio)polymers and in optimization problems.