  We study a one-dimensional lattice of $N$ sites each occupied by a
mathematical "polymer," that is, is a binary random sequence of arbitrary
length $n$, or equivalently, a rooted path of $n$ links on an infinite binary
tree. The average polymer length is controlled by the monomer fugacity $z$. A
pair of polymers on adjacent sites carries a weight factor $\omega$ for each
link on the tree that they have in common. The phase diagram in the $z\omega$
plane exhibits a critical line $z=z_{\rm c}(\omega)$. For $z<z_{\rm c}(\omega)$
there exists an equilibrium phase with, in particular, a finite average polymer
length. We investigate the equilibrium ensemble by transfer matrix and Monte
Carlo methods, paying particular attention to the vicinity of the critical
line. For $z>z_{\rm c}(\omega)$ the equilibrium is unstable and Monte Carlo
time evolution brings about a dynamical symmetry breaking which favors the
evolution of a small selection of polymers to ever greater length. While of
interest for its own sake, this model may also be relevant to the
prelife-to-life transition that has occurred during biological evolution. We
compare it to existing models of similar simplicity due to Wu and Higgs (2009,
2012) and to Chen and Nowak (2012).
