THERMODYNAMICS OF A BROWNIAN BRIDGE POLYMER MODEL IN A RANDOM ENVIRONMENT

TL;DR

This paper studies a continuum model combining a Brownian bridge and a discrete random walk as a random environment, analyzing its thermodynamic properties and energy limits relevant to protein conformation.

Contribution

It introduces a continuum version of a polymer model with a random environment, proving the existence and self-averaging of the free energy.

Findings

Thermodynamic limit of free energy exists and is self-averaging.

Explicit computation of the free energy in the thermodynamic limit.

Asymptotic behavior of the ground state energy estimated.

Abstract

We consider a directed random walk making either 0 or $+1$ moves and a Brownian bridge, independent of the walk, conditioned to arrive at point $b$ on time $T$. The Hamiltonian is defined as the sum of the square of increments of the bridge between the moments of jump of the random walk and interpreted as an energy function over the bridge connfiguration; the random walk acts as the random environment. This model provides a continuum version of a model with some relevance to protein conformation. The thermodynamic limit of the specific free energy is shown to exist and to be self-averaging, i.e. it is equal to a trivial --- explicitly computed --- random variable. An estimate of the asymptotic behaviour of the ground state energy is also obtained.