A Study of the Boltzmann Sequence-Structure Channel.

We rigorously study a channel that maps sequences from a finite alphabet to self-avoiding walks in the two-dimensional grid, inspired by a model of protein folding from statistical physics and studied empirically by biophysicists. This channel, which we call the Boltzmann sequence-structure channel, is characterized by a Boltzmann/Gibbs distribution with a free parameter corresponding to temperature. In our previous work, we verified empirically that the channel capacity appears to have a phase transition for small temperature and decays to zero for high temperature. In this paper, we make some progress toward theoretically explaining these phenomena. We first estimate the conditional entropy between the input sequence and the output fold, giving an upper bound which exhibits a phase transition with respect to temperature. Next, we formulate a class of parameter settings under which the dependence between walk energies is governed by their number of shared contacts. In this setting, we derive a lower bound on the conditional entropy. This lower bound allows us to conclude that the mutual information tends to zero in a nontrivial regime of high temperature, giving some support to the empirical fact regarding capacity. Finally, we construct an example setting of the parameters of the model for which the conditional entropy is exactly calculable and which does not exhibit a phase transition.