Francesco Unguendoli

Rigorous results on the bipartite mean-field model

We consider a bipartite mean-field model in which interaction coefficients and magnetic fields depend only on the groups the particles belong to. We rigorously compute the value of the limiting pressure per particle using tail estimation techniques. We study the phase space of the model in the symmetric regime without an external field and when the interaction coefficients within the two groups are identical. Magnetic field perturbations are considered.

Interpolating greedy and reluctant algorithms

In a standard NP-complete optimization problem, we introduce an interpolating algorithm between the quick decrease along the steepest descent direction (greedy dynamics) and a slow decrease close to the level curves (reluctant dynamics). We find that, for a fixed elapsed computer time, the best performance of the optimization is reached at a special value of the interpolation parameter, considerably improving the results of the pure cases of greedy and reluctant.

correlation inequalities for spin glass in one dimension

We prove two inequalities for the direct and truncated correlation for the nearest-neighboor one-dimensional Edwards-Anderson model with symmetric quenched disorder. The second inequality has the opposite sign of the GKS inequality of type II. In the non symmetric case with positive average we show that while the direct correlation keeps its sign the truncated one changes sign when crossing a suitable line in the parameter space. That line separates the regions satisfying the GKS second inequality and the one proved here.

Lack of monotonicity in spin glass correlation functions

We study the response of a spin glass system with respect to the rescaling of its interaction random variables and investigate numerically the behaviour of the correlation functions with respect to the volume. While for a ferromagnet the local energy correlation functions increase monotonically with the scale and, by consequence, with respect to the volume of the system we find that in a general spin glass model those monotonicities are violated.

Ground states for a class of deterministic spin models with glassy behaviour

We consider the deterministic model with glassy behaviour, recently introduced by Marinari, Parisi and Ritort, with Hamiltonian , where J is the discrete sine Fourier transform. The ground state found by these authors for N odd and 2N+1 prime is shown to become asymptotically degenerate when 2N+1 is a product of odd primes, and to disappear for N even. This last result is based on the explicit construction of a set of eigenvectors for J, obtained through its formal identity with the imaginary part of the propagator of the quantized unit symplectic matrix over the 2-torus.We consider the deterministic model with glassy behaviour, recently introduced by Marinari, Parisi and Ritort, with \ha\

Absence of glassy behaviour in the deterministic spherical and XY models

H=\sum_{i,j=1}^N J_{i,j}\sigma_i\sigma_j

Deterministic spin models with a glassy phase transition

, where

Resonant harmonic oscillators and eigenvalue multiplicity

J

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is the discrete sine Fourier transform. The ground state found by these authors for

Optimization strategies in complex systems

N