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We study the dynamics of the batch minority game, with random externalinformation, using generating functional techniques a la De Dominicis. Therelevant control parameter in this model is the ratio $\alpha=p/N$ of thenumber $p$ of possible values for the external information over the number $N$of trading agents. In the limit $N\to\infty$ we calculate the location$\alpha_c$ of the phase transition (signaling the onset of anomalous response),and solve the statics for $\alpha>\alpha_c$ exactly. The temporal correlationsin global market fluctuations turn out not to decay to zero for infinitelywidely separated times. For $\alpha<\alpha_c$ the stationary state is shown tobe non-unique. For $\alpha\to 0$ we analyse our equations in leading order in$\alpha$, and find asymptotic solutions with diverging volatility$\sigma=\order(\alpha^{-{1/2}})$ (as regularly observed in simulations), butalso asymptotic solutions with vanishing volatility$\sigma=\order(\alpha^{{1/2}})$. The former, however, are shown to emerge onlyif the agents' initial strategy valuations are below a specific critical value.Comment: 15 pages, 6 figures, uses Revtex. Replaced an old version of  volatility graph that. Rephrased and updated some reference

Generating functional analysis of the dynamics of the batch minority game with random external information