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Register a new userPossibility of a continuous phase transition in random-anisotropy magnets with ageneric random-axis distribution
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We reconsider the problem of the critical behavior of a three-dimensional $O(m)$ symmetric magnetic system in the presence of random anisotropy disorder with a generic trimodal random axis distribution. By introducing $n$ replicas to average over disorder it can be coarse-grained to a $phi^{4}$-theory with $m times n$ component order parameter and five coupling constants taken in the limit of $n to 0$. Using a field theory approach we renormalize the model to two-loop order and calculate the $beta$-functions within the $varepsilon$ expansion and directly in three dimensions. We analyze the corresponding renormalization group flows with the help of the Pade-Borel resummation technique. We show that there is no stable fixed point accessible from physical initial conditions whose existence was argued in the previous studies. This may indicate an absence of a long-range ordered phase in the presence of random anisotropy disorder with a generic random axis distribution.
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We study the kinetics of domain growth in ferromagnets with random exchange interactions. We present detailed Monte Carlo results for the nonconserved random-bond Ising model, which are consistent with power-law growth with a variable exponent. These results are interpreted in the context of disorder barriers with a logarithmic dependence on the domain size. Further, we clarify the implications of logarithmic barriers for both nonconserved and conserved domain growth.
We investigate thermodynamic phase transitions of the joint presence of spin glass (SG) and random field (RF) using a random graph model that allows us to deal with the quenched disorder. Therefore, the connectivity becomes a controllable parameter in our theory, allowing us to answer what the differences are between this description and the mean-field theory i.e., the fully connected theory. We have considered the random network random field Ising model where the spin exchange interaction as well as the RF are random variables following a Gaussian distribution. The results were found within the replica symmetric (RS) approximation, whose stability is obtained using the two-replica method. This also puts our work in the context of a broader discussion, which is the RS stability as a function of the connectivity. In particular, our results show that for small connectivity there is a region at zero temperature where the RS solution remains stable above a given value of the magnetic field no matter the strength of RF. Consequently, our results show important differences with the crossover between the RF and SG regimes predicted by the fully connected theory.
We consider an Erdos-Renyi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N->infinity one obtains a graph of finite mean degree c. In this regime, we study the distribution of resistance distances between the vertices of this graph and develop an auxiliary field representation for this quantity in the spirit of statistical field theory. Using this representation, a saddle point evaluation of the resistance distance distribution is possible at N->infinity in terms of an 1/c expansion. The leading order of this expansion captures the results of numerical simulations very well down to rather small values of c; for example, it recovers the empirical distribution at c=4 or 6 with an overlap of around 90%. At large values of c, the distribution tends to a Gaussian of mean 2/c and standard deviation sqrt{2/c^3}. At small values of c, the distribution is skewed toward larger values, as captured by our saddle point analysis, and many fine features appear in addition to the main peak, including subleading peaks that can be traced back to resistance distances between vertices of specific low degrees and the rest of the graph. We develop a more refined saddle point scheme that extracts the corresponding degree-differentiated resistance distance distributions. We then use this approach to recover analytically the most apparent of the subleading peaks that originates from vertices of degree 1. Rather intuitively, this subleading peak turns out to be a copy of the main peak, shifted by one unit of resistance distance and scaled down by the probability for a vertex to have degree 1. We comment on a possible lack of smoothness in the true N->infinity distribution suggested by the numerics.
We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdos-Renyi and regular random graphs and determine their asymptotic values for large number of colors. Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.
We study the behaviour of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully-connected Levy matrices. We derive a critical line separating localized from extended states in the case of Levy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.
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