We prove a motivic stabilization result for the cohomology
of the local systems on configuration spaces of varieties over
attached to character polynomials. Our approach interprets the stabilization as a
probabilistic phenomenon based on the asymptotic independence of certain
motivicrandom variables, and gives explicit universal formulas for the limits in terms of the
exponents of a motivic Euler product for the Kapranov zeta function. The result can
be thought of as a weak but explicit version of representation stability for the
cohomology of ordered configuration spaces. In the sequel, we find similar stability
results in spaces of smooth hypersurface sections, providing new examples to be
investigated through the lens of representation stability for symmetric, symplectic
and orthogonal groups.
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