Critical scaling of lattice polymers confined to a box without endpoint restriction

TL;DR

This paper investigates phase transitions in confined lattice polymers modeled by self-avoiding walks within a square box, developing a scaling theory, proving bounds, and verifying key exponents through Monte Carlo simulations.

Contribution

It introduces a scaling theory for confined lattice polymers without endpoint restrictions and validates it with simulations, extending understanding of phase transitions in such systems.

Findings

Identified a phase transition between empty and dense phases.

Derived bounds on the free energy of the system.

Verified key critical exponents through Monte Carlo simulations.

Abstract

We study self-avoiding walks on the square lattice restricted to a square box of side $L$ weighted by a length fugacity without restriction of their end points. This models a confined polymer in dilute solution. The model admits a phase transition between an `empty' phase, where the average length of walks are finite and the density inside large boxes goes to zero, to a `dense' phase, where there is a finite positive density. We prove various bounds on the free energy and develop a scaling theory for the phase transition based on the standard theory for unconstrained polymers. We compare this model to unrestricted walks and walks that whose endpoints are fixed at the opposite corners of a box, as well as Hamiltonian walks. We use Monte Carlo simulations to verify predicted values for three key exponents: the density exponent $\alpha=1/2$, the finite size crossover exponent $1/\nu=4/3$ and the critical partition function exponent $2-\eta=43/24$. This implies that the theoretical framework relating them to the unconstrained SAW problem is valid.