  We propose a new physically-based ``multifractal stress activation'' model of
earthquake interaction and triggering based on two simple ingredients: (i) a
seismic rupture results from activated processes giving an exponential
dependence on the local stress; (ii) the stress relaxation has a long memory.
The combination of these two effects predicts in a rather general way that
seismic decay rates after mainshocks follow the Omori law 1/t^p with exponents
p linearly increasing with the magnitude M of the mainshock and the inverse
temperature. We carefully test the prediction on the magnitude dependence of p
by a detailed analysis of earthquake sequences in the Southern California
Earthquake catalog. We find power law relaxations of seismic sequences
triggered by mainshocks with exponents p increasing with the mainshock
magnitude by approximately 0.1-0.15 for each magnitude unit increase, from
p(M=3) \approx 0.6 to p(M=7) \approx 1.1, in good agreement with the prediction
of the multifractal model. The results are robust with respect to different
time intervals, magnitude ranges and declustering methods. When applied to
synthetic catalogs generated by the ETAS (Epidemic-Type Aftershock Sequence)
model constituting a strong null hypothesis with built-in magnitude-independent
$p$-values, our procedure recovers the correct magnitude-independent p-values.
Our analysis thus suggests that a new important fact of seismicity has been
unearthed. We discuss alternative interpretations of the data and describe
other predictions of the model.
