The infinite-range-interaction Ising spin glass, in the presence of an external random field, is investigated through the replica method. At each site, the field follows a bimodal distribution, assuming the values Ϯh 0 . Within the replica-symmetry approximation, the phase diagram is obtained for different values of h 0 . The border of the ferromagnetic phase displays interesting behavior, depending on the value of h 0 , with two threshold values ͑h 0(1) and h 0 (2) ͒: ͑i͒ a continuous line, for h 0 Ͻh 0 (1) ; ͑ii͒ two pieces, one continuous ͑high temperatures͒ and another of the first-order type ͑low temperatures͒, connected at a tricritical point, for h 0 Ͼh 0 (2) ; and ͑iii͒ two continuous pieces ͑high and low temperatures͒ and a first-order part in between, with two tricritical points, for h 0 (1) рh 0 рh 0 (2) . The stability of the replica-symmetric solution is analyzed. It is shown that the higher-temperature tricritical point is always in a stable region of the phase diagram, whereas the lower-temperature one is, most of the time, inside the unstable region. Along the first-order critical line, a small gap is found between the borders associated with the instabilities of the replica-symmetric solution from either side of the phase-coexistence region, i.e., these instability lines do not meet at the ferromagnetic frontier, as usually happens in the case of second-order phase transitions. ͓S1063-651X͑98͒07705-8͔

The multifractal properties of the Edwards-Anderson order parameter of the short-range Ising spin glass model on d = 3 diamond hierarchical lattices is studied via an exact recursion procedure. The profiles of the local order parameter are calculated and analysed within a range of temperatures close to the critical point with four symmetric distributions of the coupling constants (Gaussian, Bimodal, Uniform and Exponential). Unlike the pure case, the multifractal analysis of these profiles reveals that a large spectrum of the α-Hölder exponent is required to describe the singularities of the measure defined by the normalized local order parameter, at and below the critical point. Minor changes in these spectra are observed for distinct initial distributions of coupling constants, suggesting an universal spectra behavior. For temperatures slightly above Tc, a dramatic change in the F (α) function is found, signalizing the transition. 05.50.+q, 75.10.Nr and 64.60.A I. INTRODUCTIONThe understanding of the nature of the spin-glass (SG) condensed phase in real systems has been challenging many authors [1], since the scenario emerging from Parisi's mean-field solution [2] of the SherringtonKirkpatrick (SK) [3] model came out. Some raised conclusions, like the structure of the free-energy barriers corresponding to many distinct phases below T c (pure states) [4], arranged in an ultrametric structure and the existence of a critical ordering field for the condensed phase [5], generated controversies that remain not satisfactorily elucidated. In particular, the domain-wall phenomenological scaling approach (droplet model) dismiss the SK model as appropriated for the description of short-range Ising spin glasses in low dimensions and does not share the same conclusions [6,7]. On the other hand, recent works based on numerical simulations presented results indicating that short-range models should exhibit the same qualitative features appearing in the SK model [8]. It is worth to mention that many efforts have been devoted to investigate exactly-solvable short-range SG models as an attempt to describe real spin glasses, where certain aspects of the system, e.g., the correlation length, the sensibility to the boundary conditions, and finite-size effects should present a very distinct behavior from those of infinite-range models. For the SG model on the pathological Bethe lattice with finite connectivity, the controversy about the nature of the condensed phase still persists. For instance, it was found that a replica-symmetric solution is stable for zero field when open (uncorrelated) boundary conditions are considered [9], while the breaking of replica symmetry is required to obtain a stable solution below T c when closed (correlated) boundary conditions are imposed to the system [10]. Another line of approach in the study of short-range SG behavior was developed after the work of Southern and 1 Young [11] who succeeded to obtain, by using the MigdalKadanoff renormalization group (MKRG) scheme, phase diagrams showing the pr...

The characterization of the long-range order and fractal properties of DNA sequences has proved a difficult though rewarding task due mainly to the mosaic character of DNA consisting of many interwoven patches of various lengths with different nucleotide constitutions. We apply here a recently proposed generalization of the detrended fluctuation analysis method to show that the DNA walk construction, in which the DNA sequence is viewed as a time series, exhibits a monofractal structure regardless of the existence of local trends in the series. In addition, we point out that the monofractal structure of the DNA walks carries over to an apparently alternative graphical construction given by the projection of the DNA walk into the d spatial coordinates, termed DNA trails. In particular, we calculate the fractal dimension Dt of the DNA trails using a well-known result of fractal theory linking Dt to the Hurst exponent H of the corresponding DNA walk. Comparison with estimates obtained by the standard box-counting method allows the evaluation of both finite-length and local trends effects.