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Using the ETAS branching model of triggered seismicity, we apply theformalism of generating probability functions to calculate exactly the averagedifference between the magnitude of a mainshock and the magnitude of itslargest aftershock over all generations. This average magnitude difference isfound empirically to be independent of the mainshock magnitude and equal to1.2, a universal behavior known as Bath's law. Our theory shows that Bath's lawholds only sufficiently close to the critical regime of the ETAS branchingprocess. Allowing for error bars +- 0.1 for Bath's constant value around 1.2,our exact analytical treatment of Bath's law provides new constraints on theproductivity exponent alpha and the branching ratio n: $0.9 <= alpha <= 1$ and0.8 <= n <= 1. We propose a novel method for measuring alpha based on thepredicted renormalization of the Gutenberg-Richter distribution of themagnitudes of the largest aftershock. We also introduce the ``second Bath's lawfor foreshocks: the probability that a main earthquake turns out to be theforeshock does not depend on its magnitude.Comment: 25 pages with 8 figure

Distribution of the largest aftershocks in branching models of triggered seismicity: Theory of the universal Båth law