An Ising model including field randomness and ϮJ bond randomness is studied by renormalization-group theory in spatial dimensions dϭ2 and 3. In dϭ3, with field randomness but no ϮJ randomness, ferromagnetic and paramagnetic phases occur with no intervening spin-glass phase. Also in dϭ3, at sufficient ϮJ randomness, a spin-glass phase occurs, but is replaced by the paramagnetic phase for any nonzero random field, also implying its disappearance for any nonzero uniform field. In dϭ2, no ferromagnetic phase under random fields and no spin-glass phase occur. Global phase diagrams uniting the random-field and spin-glass problems are evaluated. The strong violation of critical phenomena universality, previously found adjoining random-bond tricriticality in dϭ3, is seen here in dϭ2.

The Langevin dynamics of a self -interacting chain embedded in a quenched random medium is investigated by making use of the generating functional method and one -loop (Hartree) approximation. We have shown how this intrinsic disorder causes different dynamical regimes. Namely, within the Rouse characteristic time interval the anomalous diffusion shows up. The corresponding subdiffusional dynamical exponents have been explicitly calculated and thoroughly discussed. For the larger time interval the disorder drives the center of mass of the chain to a trap or frozen state provided that the Harris parameter, (∆/b d )N 2−νd ≥ 1, where ∆ is a disorder strength, b is a Kuhnian segment length, N is a chain length and ν is the Flory exponent. We have derived the general equation for the non -ergodicity function f (p) which characterizes the amplitude of frozen Rouse modes with an index p = 2πj/N . The numerical solution of this equation has been implemented and shown that the different Rouse modes freeze up at the same critical disorder strength ∆c ∼ N −γ where the exponent γ ≈ 0.25 and does not depend from the solvent quality.

We study the properties of polyelectrolyte chains under different solvent conditions, using a variational technique. The free energy and the conformational properties of a polyelectrolyte chain are studied by minimizing the free energy FN , depending on N (N − 1)/2 trial probabilities that characterize the conformation of the chain. The Gaussian approximation is considered for a ring of length 2 4 < N < 2 8 and for an open chain of length 50 < N < 200 in poor-and theta-solvent conditions, including a Coulomb repulsion between the monomers. In theta-solvent conditions the blob size is measured and found in agreement with scaling theory, including charge depletion effects, expected for the case of an open chain. In poor-solvent conditions, a globule instability, driven by electrostatic repulsion, is observed. We notice also inhomogeneous behavior of the monomer-monomer correlation function, reminiscence of necklace formation in poor-solvent polyelectrolyte solutions. A global phase diagram in terms of solvent quality and inverse Bjerrum length is presented. PACS. 05.20.-y Classical statistical mechanics -36.20.-r Macromolecules and polymer molecules -82.35.Rs Polyelectrolytes

A variational approach is considered to calculate the free energy and the conformational properties of a polyelectrolyte chain in d dimensions. We consider in detail the case of pure Coulombic interactions between the monomers, when screening is not present, in order to compute the end-to-end distance and the asymptotic properties of the chain as a function of the polymer chain length N . We find R ≃ N ν (log N ) γ where ν = 3 λ+2 and λ is the exponent which characterize the long-range interaction U ∝ 1/r λ . The exponent γ is shown to be non-universal, depending on the strength of the Coulomb interaction. We check our findings, by a direct numerical minimization of the variational energy for chains of increasing size 2 4 < N < 2 15 . The electrostatic blob picture, expected for small enough values of the interaction strength, is quantitatively described by the variational approach. We perform a Monte Carlo simulation for chains of length 2 4 < N < 2 10 . The non universal behavior of the exponent γ previously derived within the variational method, is also confirmed by the simulation results. Non-universal behavior is found for a polyelectrolyte chain in d = 3 dimension. Particular attention is devoted to the homopolymer chain problem, when short range contact interactions are present.

We obtain phase diagrams of regular and irregular finite-connectivity spin glasses. Contact is first established between properties of the phase diagram and the performance of low-density parity check ͑LDPC͒ codes within the replica symmetric ͑RS͒ ansatz. We then study the location of the dynamical and critical transition points of these systems within the one step replica symmetry breaking theory ͑RSB͒, extending similar calculations that have been performed in the past for the Bethe spin-glass problem. We observe that the location of the dynamical transition line does change within the RSB theory, in comparison with the results obtained in the RS case. For LDPC decoding of messages transmitted over the binary erasure channel we find, at zero temperature and rate R =1/4, an RS critical transition point at p c Ӎ 0.67 while the critical RSB transition point is located at p c Ӎ 0.7450± 0.0050, to be compared with the corresponding Shannon bound 1 − R. For the binary symmetric channel we show that the low temperature reentrant behavior of the dynamical transition line, observed within the RS ansatz, changes its location when the RSB ansatz is employed; the dynamical transition point occurs at higher values of the channel noise. Possible practical implications to improve the performance of the state-ofthe-art error correcting codes are discussed.