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Appendix C Overlap of Replicas and Replica Symmetry Ansatz
In a collective computational task, the simplest manner by which computation is accomplished refers to the associative memory problem stated as follows.
When a set of patterns {ξiμ}, labeled by , is stored in a network with N interconnected units (designated by i = 1, 2,
, N), the network responds to deliver whichever one of the stored patterns most closely resembles a new pattern ξi presented at the networks input.
The network stores a stable pattern (or a set of patterns) through the adjustment of its connection weights. That is, a set of patterns {ξiμ} is presented to the network during a training session and the connection strengths (Wij) are adjusted on a correlatory basis via a superposition of terms:

Such an adjustment calls for a minimization of energy functional when the overlap between the network configuration and the stored pattern ξi is largest. This energy functional is given by the Hamiltonian:

In the event minimization is not attained, the residual overlap between ξiμ and the other patterns gives rise to a cross-talk term. The cross-talk between different patterns on account of their random overlap would affect the recall or retrieval of a given pattern, especially when becomes of the order of N.
To quantify the random overlaps, one can consider the average free energy associated with the random binary pattern. Implicitly it refers to averaging the log(Z), but computation of <<log(Z)>> is not trivial. Therefore, log(Z) is specified by the relation:

and the corresponding averaging would involve Zn and not log(Z). The quantity Zn can be considered as the partition function of n copies or replicas of the original system. In other words:

where each replica is indicated by a superscript (replica index) on its Sis running from 1 to n.
In the conventional method of calculating <<Zn>> via saddle-point technique, the relevant order parameters (overlap parameters) derived at the saddle points can be assumed to be symmetric in respect to replica indices. That is, the saddle-point values of the order parameters do not depend on their replica indices. This is known as replica symmetry ansatz (hypothesis).
This replica symmetry method, however, works only in certain cases where reversal of limits is justified and <<Zn>> is calculated for integer n (eventually interpreted n as a real number). When it fails, a more rigorous method is pursued with replica symmetry breaking ansatz.
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