Glassy phases in random heteropolymers with correlated sequences

Mean-field behavior of the negative-weight percolation model on random regular graphs

We investigate both analytically and numerically the ensemble of minimum-weight loops in the negative-weight percolation model on random graphs with fixed connectivity and bimodal weight distribution. This allows us to study the mean-field behavior of this model. The analytical study is based on a conjectured equivalence with the problem of self-avoiding walks in a random medium. The numerical study is based on a mapping to a standard minimum-weight matching problem for which fast algorithms exist. Both approaches yield results that are in agreement on the location of the phase transition, on the value of critical exponents, and on the absence of any sizable indications of a glass phase. By these results, the previously conjectured upper critical dimension of d(u)=6 is confirmed.

Exact solution of a RNA-like polymer model on the Husimi lattice

We investigate a two-tolerant polymer model on the square Husimi lattice, which aims at describing the properties of RNA-like macromolecules. We solve the model in a numerically exact way, working out the grand-canonical phase diagram, both with and without taking into account the stacking effect. Besides a nonpolymerized phase, we observe two different polymerized phases characterized by a lower or higher density of doubly visited lattice bonds. The system exhibits three qualitatively different regimes, as a function of the monomer chemical potential. Below some T1 temperature and above some T2 temperature, the transition to the nonpolymerized phase is continuous, whereas, in the (T1,T2) temperature range, the transition is first order. In the dilute-solution limit, the high temperature regime corresponds to a swollen ("coil") state, the intermediate regime to a moderately collapsed ("molten") state, with a small fraction of paired segments, and the low temperature regime to an almost fully paired ("native") state. The molten state ends in a tricritical (Theta-like) transition at high temperature and in a critical end point at low temperature. Upon increasing the stacking energy parameter, the temperature range of the molten state turns out to be progressively reduced but never completely removed.

RNA-like polymer model: exact calculation on the Bethe lattice

We consider a lattice polymer model (random walk), in which the walk is allowed to visit lattice bonds at most twice. Such a model might have some relevance to describe statistical properties of RNA molecules. In order to mimic base pairing, we assign an attractive energy term to each doubly visited bond, and a further contribution to each pair of consecutive doubly visited bonds. The latter term is expected to mimic the stacking effect, whereas no effect of sequence, that is, of chemical specificity, is taken into account. The phase diagram is worked out exactly on a Bethe lattice, in a grand-canonical formulation. In the single molecule limit, the system undergoes two different phase transitions upon decreasing temperature: a Theta-like collapse from a swollen "coil" state to a "molten" state, with a low fraction of doubly visited bonds, and subsequently to a "paired" state, with empty or doubly visited bonds only. The stacking effect drives the latter transition from second to first order.

Threshold values, stability analysis, and high-q asymptotics for the coloring problem on random graphs

We consider the problem of coloring Erdös-Rényi and regular random graphs of finite connectivity using q colors. It has been studied so far using the cavity approach within the so-called one-step replica symmetry breaking (1RSB) ansatz. We derive a general criterion for the validity of this ansatz and, applying it to the ground state, we provide evidence that the 1RSB solution gives exact threshold values c(q) for the transition from the colorable to the uncolorable phase with q colors. We also study the asymptotic thresholds for q>>1 finding c(q) =2q ln q-ln q-1+o (1) in perfect agreement with rigorous mathematical bounds, as well as the nature of excited states, and give a global phase diagram of the problem.