arXiv subjects

R. Erichsen Jr

Publications and source records attributed to R. Erichsen Jr.

8 recordsLinked to original sources

arXiv2025

Decorated Clusters and Geometrical Frustration in Cluster Spin Glass: A Random Graph Approach

We develop a theory to investigate how geometrically frustrated clusters that become decorated affect the Cluster Spin Glass phase. The cluster structure is assumed to be a tetrahedron composed of Ising spins with z-anisotropy placed at its vertices that interact antiferromagnetically. We consider the probability $1-p_J$ of finding an impurity at a vertex of the tetrahedron that interacts ferromagnetically with the remaining elements inside the tetrahedron. An intercluster disorder is added as a random Gaussian interaction. The order parameters are obtained using the sparse random graph technique, which introduces the connectivity of the network of clusters as a controllable parameter in the theory. We examine changes that occur in the Cluster Spin Glass phase as a function of $p_J$ and $c$, in addition to the antiferromagnetic intracluster couplings $J_1$. For intermediate values of $p_J$, unexpected results appear. Even when some clusters contain a ferromagnetic impurity, there will still be robust geometric frustration effects in the cluster network. However, the $p_J$ threshold for this to occur depends on connectivity. Conversely, below this threshold, reduced GF effects favor the reappearance of the CSG phase. Furthermore, the Curie-Weiss temperature $\Theta_W$ has a gradual change of signal, indicating that the effects of the impurities extend to the paramagnetic phase.

cond-mat.stat-mech↗

arXiv2021

Ising spin glass in a random network with a gaussian random field

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.

cond-mat.dis-nn↗

arXiv2019

Unfolding of phases and multicritical points in the Classical Anisotropic van Hemmen Spin Glass Model with Random Field

We study magnetic properties of the 3-state spin ($S_{i}=0$ and $\pm 1$) spin glass (SG) van Hemmen model with ferromagnetic interaction $J_0$ under a random field (RF). The RF follows a bimodal distribution The combined effect of the crystal field $D$ and the special type of on-site random interaction of the van Hemmen model engenders the unfolding of the SG phases for strong enough RF, i. e., instead of one SG phase, we found two SG phases. Moreover, as $J_0$ is finite, there is also the unfolding of the mixed phase (with the SG order parameter and the spontaneous magnetization simultaneously finite) in four distinct phases. The emergence of these new phases separated by first and second order line transitions produces a multiplication of triple and multicritical points.

cond-mat.dis-nn↗

arXiv2005

Synchronous versus sequential updating in the three-state Ising neural network with variable dilution

The three-state Ising neural network with synchronous updating and variable dilution is discussed starting from the appropriate Hamiltonians. The thermodynamic and retrieval properties are examined using replica mean-field theory. Capacity-temperature phase diagrams are derived for several values of the pattern activity and different gradations of dilution, and the information content is calculated. The results are compared with those for sequential updating. The effect of self-coupling is established. Also the dynamics is studied using the generating function technique for both synchronous and sequential updating. Typical flow diagrams for the overlap order parameter are presented. The differences with the signal-to-noise approach are outlined.

cond-mat.dis-nn↗

arXiv2004

The three-state layered neural network with finite dilution

The dynamics and the stationary states of an exactly solvable three-state layered feed-forward neural network model with asymmetric synaptic connections, finite dilution and low pattern activity are studied in extension of a recent work on a recurrent network. Detailed phase diagrams are obtained for the stationary states and for the time evolution of the retrieval overlap with a single pattern. It is shown that the network develops instabilities for low thresholds and that there is a gradual improvement in network performance with increasing threshold up to an optimal stage. The robustness to synaptic noise is checked and the effects of dilution and of variable threshold on the information content of the network are also established.

cond-mat.dis-nn↗

arXiv2002

Retrieval and Chaos in Extremely Diluted Non-Monotonic Neural Networks

We discuss, in this paper, the dynamical properties of extremely diluted, non-monotonic neural networks. Assuming parallel updating and the Hebb prescription for the synaptic connections, a flow equation for the macroscopic overlap is derived. A rich dynamical phase diagram was obtained, showing a stable retrieval phase, as well as a cycle two and chaotic behavior. Numerical simulations were performed, showing good agreement with analytical results. Furthermore, the simulations give an additional insight into the microscopic dynamical behavior during the chaotic phase. It is shown that the freezing of individual neuron states is related to the structure of chaotic attractors.

cond-mat.dis-nn↗

arXiv2001

Retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks

The retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks are derived and studied in replica-symmetric mean-field theory generalizing earlier works on either the fully connected or the symmetrical extremely diluted network. Capacity-gain parameter phase diagrams are obtained for the Q=3, Q=4 and $Q=\infty$ state networks with uniformly distributed patterns of low activity in order to search for the effects of a gradual dilution of the synapses. It is shown that enlarged regions of continuous changeover into a region of optimal performance are obtained for finite stochastic noise and small but finite connectivity. The de Almeida-Thouless lines of stability are obtained for arbitrary connectivity, and the resulting phase diagrams are used to draw conclusions on the behavior of symmetrically diluted networks with other pattern distributions of either high or low activity.

cond-mat.dis-nn↗

arXiv1998

A canonical ensemble approach to graded-response perceptrons

Perceptrons with graded input-output relations and a limited output precision are studied within the Gardner-Derrida canonical ensemble approach. Soft non- negative error measures are introduced allowing for extended retrieval properties. In particular, the performance of these systems for a linear and quadratic error measure, corresponding to the perceptron respectively the adaline learning algorithm, is compared with the performance for a rigid error measure, simply counting the number of errors. Replica-symmetry-breaking effects are evaluated.

cond-mat.dis-nn↗