We study the spin-glass on a Bethe lattice using replicas. The problem is to find the fixed point of an iterative map in a 2n-dimensional space as n approaches zero. We show that in the high-temperature phase the fixed point is replica symmetric, but in the spin-glass phase this becomes unstable. Making the Parisi ansatz we show that the overlap function P(q) has the same form as in the SK model.

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We present the results of analytical and numerical calculations for the zero temperature parallel dynamics of spin glass and neural network models. We use an analytical approach to calculate the magnetization and the overlaps after a few time steps. For the long time behaviour, the analytical approach becomes too complicated and we use numerical simulations. For the Sherrington-Kirkpatrick model, we measure the remanent magnetization and the overlaps at different times and we observe power law decays towards the infinite time limit. When one iterates two configurations in parallel, their distance d(∞) in the limit of infinite time depends on their initial distance d(0). Our numerical results suggest that d(∞) has a finite limit when d(0) → 0. This result can be regarded as a collective effect between an infinite number of spins. For the Little-Hopfield model, we compute the time evolution of the overlap with a stored pattern. We find regimes for which the system learns better after a few time steps than in the infinite time limit

PACS 02.50.-r -Probability theory, stochastic processes and statistics PACS 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion PACS 65.60.+a -Thermal properties of amorphous solids and glasses: heat capacity, thermal expansion, etc.Abstract -It is well known that the mean field theory of directed polymers in a random medium exhibits replica symmetry breaking with a distribution of overlaps which consists of two delta functions. Here we show that the leading finite size correction to this distribution of overlaps has a universal character which can be computed explicitly. Our results can also be interpreted as genealogical properties of branching Brownian motion or of branching random walks.Introduction. -The study of branching Brownian motion and of branching random walks is central in the theory of probability [1-4] and appears in several physical contexts [5][6][7][8][9][10]. Here we focus on its revelance in the mean field theory of directed polymers in a random medium [11][12][13][14]. This mean field version is an example of a disordered system which exhibits replica symmetry breaking in its low temperature phase, [15][16][17]. In the models considered here only one step of replica symmetry breaking is required in the thermodynamic limit [18]. Our motivation here is to determine how this broken symmetry is affected by finite size fluctuations. In the present work we will show that the one step replica symmetry breaking is smoothed in a universal way for which one can obtain an explicit analytic expression.

We present a systematict and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as "complex replica numbers". Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the "complex replica numbers" near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally our approach allows one to see why some apparently diverging series or integrals are harmless.