Complete condensation in a zero range process on scale-free networks.

We study a zero range process on scale-free networks in order to investigate how network structure influences particle dynamics. The zero range process is defined with the rate p(n) = n(delta) at which particles hop out of nodes with n particles. We show analytically that a complete condensation occurs when delta < or = delta(c) triple bond 1/(gamma-1) where gamma is the degree distribution exponent of the underlying networks. In the complete condensation, those nodes whose degree is higher than a threshold are occupied by macroscopic numbers of particles, while the other nodes are occupied by negligible numbers of particles. We also show numerically that the relaxation time follows a power-law scaling tau approximately L(z) with the network size L and a dynamic exponent z in the condensed phase.

Analytic study of the three-urn model for separation of sand.

We present an analytic study of the three-urn model for separation of sand, which can be regarded as a zero-range process. We solve analytically the master equation and the first-passage problem. We find that the stationary probability distribution obeys the detailed balance and is governed by the free energy. We find that the characteristic lifetime of a cluster diverges algebraically with exponent 1/3 at the limit of stability. We also give a general argument that the scaling behavior is robust with respect to different expressions of the flux.

Optimal capacity of the Blume-Emery-Griffiths perceptron.

A Blume-Emery-Griffiths perceptron model is introduced and its optimal capacity is calculated within the replica-symmetric Gardner approach, as a function of the pattern activity and the embedding stability parameter. The stability of the replica-symmetric approximation is studied via the analog of the de Almeida-Thouless line. A comparison is made with other three-state perceptrons.

Analytic study of the urn model for separation of sand.

We present an analytic study of the urn model for separation of sand recently introduced by Lipowski and Droz [Phys. Rev. E 65, 031307 (2002)]. We solve analytically the master equation and the first-passage problem. The analytic results confirm the numerical results obtained by Lipowski and Droz. We find that the stationary probability distribution and the shortest one among the characteristic times are governed by the same free energy. We also analytically derive the form of the critical probability distribution on the critical line, which supports their results obtained by numerically calculating Binder cumulants (A. Lipowski and M. Droz, e-print cond-mat/0201472).

Gap structure of the local field in symmetric Q Ising neural networks.

The time evolution of the local field in symmetric Q-Ising neural networks is studied for arbitrary Q. In particular, the structure of the noise and the appearance of gaps in the probability distribution are discussed. Results are presented for several values of Q and compared with numerical simulations.

Optimal nonlinear training in the multi-class proximity problem.

Using a signal-to-noise analysis, the effects of nonlinear modulation of the Hebbian learning rule in the multi-class proximity problem are investigated. Both random classification and classification provided by a Gaussian and a binary teacher are treated. Analytic expressions are derived for the learning and generalization rates around an old and a new prototype. For the proximity problem with binary inputs but Q'-state outputs, it is shown that the optimal modulation is a combination of a hyperbolic tangent and a linear function. As an illustration, numerical results are presented for the two-class and the Q' = 3 multi-class problem.