Parallel dynamics of disordered Ising spin systems on finitely connected directed random graphs with arbitrary degree distributions

The dynamics of non-equilibrium Ising model with parallel updates is investigated using a generalized mean field approximation that incorporates multiple two-site correlations at any two time steps, which can be obtained recursively. The proposed method shows significant improvement in predicting local system properties compared to other mean field approximation techniques, particularly in systems with symmetric interactions. Results are also evaluated against those obtained ...

We characterize the generating function of bipartite planar maps counted according to the degree distribution of their black and white vertices. This result is applied to the solution of the hard particle and Ising models on random planar lattices. We thus recover and extend some results previously obtained by means of matrix integrals. Proofs are purely combinatorial and rely on the idea that planar maps are conjugacy classes of trees. In particular, these trees explain wh...

We study the stochastic dynamics of Ising spin models with random bonds, interacting on finitely connected Poissonnian random graphs. We use the dynamical replica method to derive closed dynamical equations for the joint spin-field probability distribution, and solve these within the replica symmetry ansatz. Although the theory is developed in a general setting, with a view to future applications in various other fields, in this paper we apply it mainly to the dynamics of the...

This thesis focus on the extension of the Parisi full replica symmetry breaking solution to the Ising spin glass on a random regular graph. We propose a new martingale approach, that overcomes the limits of the Parisi-M\'ezard cavity method, providing a well-defined formulation of the full replica symmetry breaking problem in random regular graphs. We obtain a variational free energy functional, defined by the sum of two variational functionals (auxiliary variational function...

We present several efficient implementations of the simulated annealing algorithm for Ising spin glasses on sparse graphs. In particular, we provide a generic code for any choice of couplings, an optimized code for bipartite graphs, and highly optimized implementations using multi-spin coding for graphs with small maximum degree and discrete couplings with a finite range. The latter codes achieve up to 50 spin flips per nanosecond on modern Intel CPUs. We also compare the per...

We present exact expressions for hysteresis loops in the ferromagnetic random field Ising model in the limit of zero temperature and zero driving frequency for an arbitrary initial state of the model on a Bethe lattice. This work extends earlier results that were restricted to an initial state with all spins pointing parallel to each other.

In this paper, we extend the full replica symmetry breaking scheme to the Ising spin glass on a random regular graph. We propose a new martingale approach, that overcomes the limits of the Parisi-M\'ezard cavity method, providing a well-defined formulation of the full replica symmetry breaking problem in random regular graphs. Finally, we define the order parameters of the system and get a set of self-consistency equations for the order parameters and the free energy. We face...

We propose a statistical mechanics approach to a coevolving spin system with an adaptive network of interactions. The dynamics of node states and network connections is driven by both spin configuration and network topology. We consider a Hamiltonian that merges the classical Ising model and the statistical theory of correlated random networks. As a result, we obtain rich phase diagrams with different phase transitions both in the state of nodes and in the graph topology. We ...

We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g. energy) and include states with diverging correlation length and unbounded multi-particle entanglement. As a demonstration we apply our approach to the Ising model on ...

We investigate a kinetic Ising model with several single-spin flip dynamics (including Metropolis and heat-bath) on quenched and annealed random regular graphs. As expected, on the quenched structures all proposed algorithms reproduce the same results since the conditions for the detailed balance and the Boltzmann distribution in an equilibrium are satisfied. However, on the annealed graphs situation is far less clear -- the network annealing disturbs the equilibrium moving t...