Bernard Derrida

Random Networks of Automata: A Simple Annealed Approximation

Kauffmans model is a random complex automata where nodes are randomly assembled. Each node σi receives K inputs from K randomly chosen nodes and the values of σi at time t + 1 is a random Boolean function of the K inputs at time t. Numerical simulations have shown that the behaviour of this model is very different for K > 2 and K ≤ 2. It is the purpose of this work to give a simple annealed approximation which predicts K = 2 as the critical value of K. This approximation gives also quantitative predictions for distances between iterated configurations. These predictions agree rather well with the numerical simulations. A possible way of improving this annealed approximation is proposed.

An Exactly Solvable Asymmetric Neural Network Model

We consider a diluted and nonsymmetric version of the Little-Hopfield model which can be solved exactly. We obtain the analytic expression of the evolution of one configuration having a finite overlap on one stored pattern. We show that even when the system remembers, two different configurations which remain close to the same pattern never become identical. Lastly, we show that when two stored patterns are correlated, there exists a regime for which the system remembers these patterns without being able to distinguish them.

Shift in the velocity of a front due to a cutoff

We consider the effect of a small cut-off epsilon on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cut-off epsilon is to select a single velocity which converges when epsilon tends to 0 to the one predicted by the marginal stability argument. For small epsilon, the shift in velocity has the form K(log epsilon)^(-2) and our prediction for the constant K agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.

Polymers on disordered trees, spin glasses, and traveling waves

We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions that move at all possible speeds above a certain minimal speed. The speed of the wavefront is the free energy of the polymer problem and the minimal speed corresponds to a phase transition to a glassy phase similar to the spin-glass phase. Several properties of the polymer problem can be extracted from the correspondence with the traveling wave: probability distribution of the free energy, overlaps, etc.

Optimal storage properties of neural network models

The authors calculate the number, p= alpha N of random N-bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter K. For each value of alpha and K, there is a minimum fraction fmin of wrong bits. They find a critical line, alpha c(K) with alpha c(0)=2. The minimum fraction of wrong bits vanishes for alpha alpha c(K). The calculations are done using a saddle-point method and the order parameters at the saddle point are assumed to be replica symmetric. This solution is locally stable in a finite region of the K, alpha plane including the line, alpha c(K) but there is a line above which the solution becomes unstable and replica symmetry must be broken.

Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process

AbstractWe consider an open one dimensional lattice gas on sites i=1,..., N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when N→∞. The probability of microscopic configurations corresponding to some other profile ρ(x), x=i/N, has the asymptotic form exp[−N

Phase Transitions in Two-Dimensional Kauffman Cellular Automata

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Fractal structure of zeros in hierarchical models

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